There are various kinds of fermentation, which are named fully fastened to it in such a way that their points nearly touch after their products, as alcoholic, acetic, butyric fermentations. one another. These spangles are usually cut with sides about It seems that the substance of the ferment does not mingle three-fourths of an inch long, and a hole is punched out of the with the products; but the plant or animal is propagated at centre of each, so that the sparks between the points of those the expense of the albumen, or other nitrogenous matter, which on the interior are seen through the apertures in those outside. may be present; and, as might be anticipated, the substance of The annexed sketch will render this more clear; the shaded parts the ferment is nitrogenous matter. in it represent the clear glass (Fig. 14). As will be seen shortly, In what way the ferment determines the decomposition of the a plane sheet of glass may be used for this instead of a jar, and compounds in its neighbourhood is at present a mystery. it is much easier to fasten the spangles evenly to it. Considerable Alcoholic Fermentation.-When the juices of fruits, etc., which care is required in placing them, but the beauty of the expericontain sugar are kept at a temperature of about 20° Cent. for ment will repay the trouble. some hours, they become turbid and begin to work, or ferment. As we saw in the last lesson, the opposite coatings of the jar Bubbles of gas escape, which is carbonic acid. When once act upon each other by induction; the charge, however, is this change has fairly set in, the presence of air is no longer lodged on the surfaces of the glass , and not in the tinfoil, as necessary. When the gas has ceased to come off, upon dis- might at first be supposed. The foil merely serves to convey tilling the liquor, alcohol is found to come over at the commence the electricity, and distribute it over the surface of the glass. ment of the operation. If the liquid bo examined after the An experimental proof of this can easily be furnished. Procure fermentation is complete, it is found to contain yeast , which, a jar of a somewhat conical form, as shown at B (Fig. 15), and ander the microscope, appears to be oval organised bodies, about let c be a tin or zinc covering made just to fit it. Procure also , B, propagate by budding-after the manner of certain cacti . The a knob by which it may be charged fixed to it. Now put the growth of this fungus is prohibited if any of the mineral acids whole together, as at a, and we shall have a complete Leyden be present, if the solution contain more than a quarter of its jar; the only difference between it and those we have been con. weight of sugar, or twenty per cent. of alcohol. The presence sidering being that the coatings are movable, instead of being of many of the metallio salts also arrests fermentation. There fixed to the glass. seem to be two varieties of yeast oberhefe (surface yeast) and This jar may now be charged in the usual way, and then two. It appears in detached cells, which do not propagate by or gutta-percha. The object of this is to prevent the shock being bads, as the oberhofe, but by spores which the larger cells received when the knob is touched, for if the jar be placed on a throw off. table as usual, the wood, etc., will complete the circuit through The Rising of Bread. - In the making of bread, the flour the person and the floor, and a violent shock will be felt; the dough is mixed with yeast- either fresh brewer's yeast, or non-conductor, however, entirely prevents this. Having thus German yeast, which is yeast dried by pressure — a process insulated the jar, we may remove the interior cylinder by means which does not injure the vitality of the plant. Some of the of the rod, and, on testing it, we shall detect very little free starch of the flour acted upon by the yeast is converted into electricity in it. Now carefully lift the glass by its edges out of sugne. This undergoes fermentation. The alcohol escapes into the outer coating; this too, like the inner one, will be found to the oven, and the carbonio acid being liberated from all points be neutral. The proof plane, however, if applied to the inner in the dough, blows up the bread by filling it with bubbles of or outer side of the glass, will at once show the presence of the gas, the walls of the holes being rendered tenacious by electricity, and on restoring the coatings the jar may be disthe gluten. Bread may also be "raised” by "baking powder," charged as at first. If a second set of coatings be made, the or by a mixture of sodio carbonate and hydrochloric acid, the jar may be placed in them, and will be found to have lost but result being common salt, a necessary ingredient of good bread, little of its charge. and carbonic acid, which makes the bread spongy. If the glass be thick, the electricity does not all remain upon Dr. Dauglish introduced a third method, which is used to make the surface, but penetrates a slight distance into its substance. "aërated bread.” The floor is mixed in a closed iron vessel This is seen by what is called the residuary charge. After a with water, which is impregnated with carbonic acid gas at a Leyden jar has been discharged, and allowed to stand a little pressnre of 100 pounds on the square inch. The dough is while, a second charge much weaker than the first may be taken drawn off into tins, and is baked in the usual way. from it, and sometimes even a third and fourth have been felt. New and Stale Bread. It is a popular fallacy that bread If the jar has been highly charged, and kept so for a short time, becomes stale because it dries. This, however, is not true, for these residuary charges are stronger. The electricity appears in both cases a loaf contains as much as forty-five per cent. of to have penetrated a little way into the glass, and therefore the water. The soft texture of new bread seems to depend on a equilibrium is not completely restored at the first discharge. certain molecular arrangement, which may be restored to stale As soon, however, as this discharge is effected, the electricity bread by re-heating it in the oven. To prevent too great har-gradually finds its way to the coating, thereby causing the dening of the crust, the loaf should be steeped for a few minutes second shock. in milk and water before being introduced into the oven a It is, however, only with a thick jar that this is much noticed, second tirse. and such a jar does not take a very high charge ; good jar should therefore be thin, and this effect should not be seen to any great degree. ELECTRICITY.--V. When filings are used instead of tinfoil for lining the inside SPOTTED JAR-JAR WITH MOVABLE COATINGS--LEYDEN PANE however, in this case is the want of absolute contact between of a jar, these secondary shocks are much stronger; the cause, - ELECTRIC PENDULUM-BATTERY --ELECTROMETERS- the particles, which prevents the electricity from passing as HARRIS'S UNIT JAR-EFFECTS OF SHOCK. rapidly as it otherwise would. INSTEAD of coating the outside of the Leyden jar with tinfoil, Now the effect of induction is not affected by the shape of it may be varnishod, and some metallic powder—as, for instance, the interposed di-electric, and hence we may obtain similar fine brass filings-sprinkled on while it is wet, a narrow band of effects if we employ a sheet of glass instead of a jar. This is tinfoil being placed round the top. The exterior will then be easily seen by taking a pane of glass, and coating both sides of illuminated as the jar is charged and discharged. To show this it evenly, leaving a clear margin of two or three inches all the to the greatest advantage, the knob should be bent over so as to way round. From this we shall be able to obtain a shock come within striking distance of the exterior coating, and the similar to that obtained from the Leyden jar. One side must jar suspended by the loop thus formed from the conductor, a be held to the conductor of the machine, and the finger placed connection being made from the exterior to the ground, as against the other, or some other plan arranged to connect it shown in Fig. 13. with the earth, and thus allow of the escape of its electricity. The piece of apparatus known as the diamond, or spotted jar, Usually, a strip of the foil is brought from the under surface to likewise affords a very interesting experiment. the edge, so as to render it more convenient to use; then the In this the glass jar, instead of being covered with a continuous pane may be laid down, and a wire or ehain from the conductor surface of tinfoil, has a number of diamond-shaped pieces care- laid upon its upper coating. If we examine the state of the А two coatings of a charged pane, we shall find that there is a i Various batteries of great size and power have thas been small amount of positive electricity on one side un-neutralised made at different times : for all ordinary purposes, however, by induction, and therefore free. The amount of this depends from four to nine jars, holding about five or six pints exch, will on the thickness of the glass. be amply sufficient. If then we hold the finger to this side, taking care that the It is important to be able to estimate with some degree of other coating is insulated, we shall obtain a small spark from it. accuracy the intensity of the charge in a jar, and different As soon as this has passed, a similar amount of negative elec- means of attaining this end have been devised. The simplest tricity will be set free at the other side, and we can then draw a is by means of the quadrant electrometer A (Fig. 17), which is spark from that; in this way, we may continue drawing off the shown on one of the knobs of the battery. It consists of a charge, drop by drop, as it were, tiil it is all gone. If a Leyden thick brass rod, surmounted by a knob, and bearing on one side jar be placed on an insulating stand, we may discharge it in a a semi-circular graduated scale, usually made of bone or ivory. simila: way. At the centre of this is suspended a thin wood rod, carrying & Two interesting experiments may be tried as illustrations of pith-ball on its further end. As the charge increases, this ball this-one with the pane, the other with the jar. Make two feet becomes more and more repelled, and the angle shown on the of gutta-percha, or some insulating material, so that the pane graduated arc increases, thereby giving a rough indication of may stand vertically. Having charged it, take a piece of wire the intensity of the charge. It must be remembered, however, and place a pith-ball on each end; then bend it almost round, that it is the intensity, and not the quantity, of the electricity 80 that when it rests on the upper edge of the pane the balls that is indicated. If the same amount of electricity be distrimay each be about an inch from the coating. The side on buted over jars or batteries having double the amount of coated which the excess exists will first attract the ball nearest to it, surface, the electrometer will only show one-half the intensity. and thus part with its excess of electricity; the other ball will still, the instrument is very useful, especially if it is made with then be attracted by the rod somewhat the other side, and smaller at the lower in this way the wire Fig. 16. end, so that it will will rock backwards fit into the prime conand forwards till the ductor or any other charge is dissipated. piece of apparatus. This is called the A better way of electric pendulum. regulating the intenTo show the same thing with a Leyden Fig. 13. sity of a shock is by means of Lane's disjar, we must fix a charging electromebell on the wire just ter, which is reprebelow the ball, A sented in Fig. 18. metal support, It depends for its (Fig. 16), carrying a action on the fact similar bell at the Fig. 14. that the distance same height, must through which a be fixed on a stand, charge will dart beand connected with B tween the balls, or, as the outer coating of Fig. 15. D it is called, the strikthe jar by a piece of ing distance, is, for wire or tinfoil, B. small charges, directThe upper part of ly proportional to this support has a the intensity of the bent wire fixed to it, charge; that is, if from which a small double the amount of ball is suspended electricity be present by a thread of silk, on the same surface If now the jar be the striking distance charged, this ball will be twice as great. will be alternately attracted and repelled by the bells, and | This distance varies, too, inversely as the amount of surface over thus will continue to ring them till the jar is discharged. which the charge is distributed. Thus, is one jar has twice the If a jar be charged, and allowed to stand, the electricity in it surface of another, and the same amount of electricity be passed will be slowly dissipated, chiefly owing to the moisture in the into each, the striking distance of the first will be only one-half air, which acts as an imperfect conductor. This may be partly that of the second. These results will easily be understood if we obviated by coating the surface of the glass with shellac recollect that it is always intensity, and not quantity, that is varnish, and thus hindering the deposit of damp which usually shown; just as a thermometer shows the intensity and not tho forms on it. When, however, it is desired to preserve a charge quantity of heat, for there clearly is a larger quantity in a gallon for any length of time, the construction of the jar is slightly of water at the temperature of the body than in a cupful at the altered. The rod is not fixed to the cap, but passes through a boiling-point, and yet the thermometer shows a much higher glass tube fixed in it to the bottom; and when the jar is temperature if immersed in the latter. charged, it may be inverted, and the rod allowed to fall out. The electrometer consists of a metal tube, A, which fits on to When the charge is required, the rod must be carefully dropped the rod of a jar or battery, and carries at one side a curved rod in again. In doing this, however, it is advisable to take the pre- of glass, B. At the end of this is fixed a second tube, c, with a caution of placing it first upon some non-conducting substance. brass wire, terminated at one end in a ball, D, and at the other Theoretically, there is no limit to the size of the jar that may in a ring, passing through it. When this is attached to a jar, the be employed. In practice, however, many inconveniences attach ball d may be brought within any desired distance of the knob, to the use of those which are very large. The tension of the and a chain is then attached to the ring and connected with the electricity frequently becomes so great in them, that if there be outside. When the jar is sufficiently charged, the electricity a flaw or thin place in the glass it will pierce it; and if it be darts between the balls and discharges it. If it be desired to made thick, to guard against this danger, the induction is con- pass this shock through the body, or through any substance, the siderably weakened. The plan therefore adopted is to employ a two sides of it are connected by brass chain or wire with the ring number of small jars, and connect them together so as to form and onter coating respectively, and the shock passes as before. a battery. They are usually placed in a tray lined with metal, There are one or two other forms of discharging electrometer so as to connect their exterior coatings, and a wire is brought which are sometimes used, and which depend on the attraction from this to one of the handles; the knobs are also connected of the knob of the jar for a balanced metal rod. We need together by wires passing through them. (Fig. 17.) not, however, stop to explain them here. one. In all these the shock is given as soon as the necessary charge reference to Fig. 19, which represents the charge from a battery is communicated; this is sometimes rather a disadvantage, and being passed through a bird by means of this discharger. may be obviated by the use of a unit jar, which consists of a A piece of hard wood is taken, about twenty inches long small Leyden jar exposing about six inches of coated surface on and six inches wide, and in the middle of it there is fixed a each side. This is supported on an insulating stand, so that its small table, which can be adjusted at a convenient height by exterior is in contact with the knob of the jar to be charged. means of a thumb-screw. The top of this table is usually made A rod and ball are also connected with its outside coating, and of a disc of glass fixed on by shellac, or else it has a strip of brought within striking distance of its inner rod. The jar is ivory inlaid across it. At each end of the board is fixed a glass now brought to the conductor of the machine, and as it becomes rod, carrying at the top a revolving cap fitted with a compass filled a corresponding amount of electricity passes from its joint, so that the wire passing through it may be inclined at any exterior into the large jar. This continues till the unit jar is | angle. The wires can also be slipped backwards or forwards, so fully charged, when it that they may easily discharges itself, and be brought into conthe same process is re tact with any part of peated. In this way, the substance to be by counting the num operated upon. These ber of discharges of B wires are pointed at the small jar, we as. the ends, and brass certain the number of balls are fitted so as units of electricity con to screw on over the tained in the large points when equired; but as brass balls are As we now clearly rather expensive, lead understand the con bullets may, in most. struction and action of Fig. 18. cases, be substituted the Leyden jar and for then, the main dif. battery, we may notice ference being in the some of the effects pro Fig. 17. appearance. One of duced by the electric these instruments will shock. We will look be found to be of great at the physiological use to the student, and effects; and the first he should therefore enwhich we observe is deavour to procure, or, the peculiar sensation better still, to make experienced when the one; for a better know charge is allowed to ledge of the principles pass through the body. of a science may usuThe sensation varies, ally be obtained by of course, with the in. making apparatus than tensity of the shock, by merely experimentand is most strongly ing with that already felt at the elbows and Fig. 19. made. As shown in across the breast. If the figure, the object the shock of a large through which the battery be taken dan. charge is to be passed gerous results may is placed on the table, ensue, especially if and the knobs or points the electric fluid pass of the wires brought through any vital por into contact with oppotion; but with ordi site sides of it. One nary charges no ill wire is then connected effects whatever seem with the outside of the to be produced ; in Fig. 20. jar or battery, and the some cases it even ap other connected with pears to be beneficial. the knob by means of The effect of a very the jointed dischargstrong charge is seen ing rod, contact being when a person is struck made with the wire of by lightning, death the discharger first. being frequently For this purpose, howcaused by it. The ac ever, it is necessary tion appears to be that it should have an mainly on the nervous system. If we pass the shock through | insulating handle, as the electricity always chooses the best small animals, such as birds or mice, they will be killed instan- conductors, and if there be any distance between the knobs, or taneously, and larger animals have been killed by shocks from if a badly conducting substance be interposed, the electricity may powerful batteries. There is, however, nothing further to be travel through the body of the operator to the ground instead learnt by these experiments, and it is therefore cruel and need of passing through the discharger less to repeat them. The next class of effects we notice are the luminous., As. The electric fluid, in its passage through living or dead bodies, already seen, whenever the spark traverses air a line of light also causes convulsive contractions of the muscles. Its physio is seen ; when the two electricities from the opposite sides of logical effects will, however, be seen much more clearly when we a jar unite, the spark is much more luminous, being thicker come to treat of Voltaic Electricity, and we may therefore leave and denser, because of the passage of a much greater amount further notice of the matter till then of the fluid. This spark is almost instantaneous, so that if a In studying the effects produced by electricity, we need some dark room be suddenly illuminated by it, a rapidly revolving method of conveniently holding an object while the shock is wheel in it will appear to be at rest. So instantaneous is it, passed through it, and this is afforded us by a very useful piece that a printed bill has been fixed on to a wheel, and made to of apparatus, known as Henley's Universal Discharger. The revolve several hundred times per minute, and yet a distinct construction and mode of using this will be understood by photograph of it has been taken by means of the light from y = 12. and y. the spark, the time required for its passage being so short that Transposing y in the 1st equation, gives a = 36 - y. the light has faded before the wheel has moved through an Transposing y in the 2nd equation, x= 12+y. appreciable space. Making these values of & equal, 12+y= 36 - y. if the points be made nearly to touch, and the thumb Transposing, etc., pressed down upon them, it will appear to be rendered semi Substituting the value of y, x=12+12= 24. transparent, and the shook will not be felt. In a similar way Hence, 24 and 12 are the values required. the shock may be passed through eggs, oranges, etc., illumi. nating them. Some substances, too, remain luminous for some EXERCISE 36. time after the charge has passed through them. 1. Given 2x + 3y = 28, and 3x + 2y = 27; to find the valaes of x and y. If we fix a ball to the plate of an air-pump, and connect it 2. Given 4x + y = 43, and 5x + 2y = 56; to find the values of x and y. with the exterior of a jar, and then place over it an open re 3. Given 4x-2y = 16, and 6x = 9y; to find the values of x and y. ceiver with a similar ball and rod passing air-tight through a 4. Given 41---2y = 20, and 4x + 2y = 100; to find the values of : plate at its upper end, we shall find that the spark will pass & much greater distance. In a long exhausted tube it will some- 5. Given 5x + 8 = 7y, and 5y + 32 = 72; to find the values of x and times appear to dart along like a ball of light. EXAMPLE (1).-To find two numbers such that their sum Similarly, if we exhaust a large globe having a wire and ball shall be 24; and the greater shall be equal to five times the less. fixed tightly in each end, and connect the upper one B (Fig. 20) Here, let æ be the greater; and y the less. with the prime conductor, we shall find that, instead of the elec Then, x+y= 24, tricity passing along in a series of thin sparks, it will spread out And D=5y. and form a large faintly luminous space, having a violet tinge. Whence, 5y +y=by = 24, On again gradually admitting the air by means of the stopcock, And this space will grow smaller and smaller, till at last the narrow Therefore, 20. Ans. 20 and 4. but bright sparks pass as before. Many similar experiments of great beauty may be performed by means of voltaic eleo EXAMPLE (2).- Find two quantities whose sum is equal to h; tricity, and it is believed that th beautiful phenomena of and the difference of whose squares is equal to d. the aurora borealis, or northern lights, so frequently seen in Letæ and y be the two quantities. high latitudes, may be attributed to a similar cause. And ** =a} per question. From the first equation we have, by transposition, LESSONS IN ALGEBRA.-XXII. a = h - Y, And, by squaring both sides, we have, aa = h - 2hy + ym. In our former Lessons on Simple Equations we gave the rules From the second equation, we have, by transposition, for solving those which contain only one unknown quantity; 202 = ga + d. and, with the exception of one or two, the whole Centenary of Now, by equating the two values of **, we have, Problems were solved by means of these rules. We proceed now to show how to resolve equations which contain two un yo+d=ha — 2hy + ya ; known quantities. And, by transposition and cancelling, we have, Cases indeed frequently occur in which two unknown quan 2hy=ha - d; tities are necessarily introduced into the same calculation. h* -d Whence, y= EXAMPLE.-Suppose the following equations are given, 2h h-dha +d (1.) «+y=14, Therefore, z=h 21. (2.) -y=2. EXAMPLE (3).-Given ax + by = h, and < +y=d; to find Hero, if y be transposed in each, they will become the values of cand y. (1.) 2 = 14 - Y, Here, from the first equation, we have, by transposition, (2.) a= 2 + y. ax=h--by, Now, the first member of each of the equations is æ, and the second member of each is equal to 3. But according to the And a= h-by axiom that quantities which are respectively equal to another quantity, are equal to each other ; therefore we have Again, from the second equation, we have, by transposition, a=d—y, 2+y=14 - Y; whence y=6. Lastly, by substituting the value of y in the 1st equation, we h- by Whence, =dy; have a +6=14; and a = 8. Therefore, 8 and 6 are the values of a and y. Or, h—by = ad- ay, In solving the preceding problem, it will be observed that And ay-by=ad - h. we first found the value of the unknown quantity æ in each From this equation, by separating the left-hand member into equation ; and then, by making one of the expressions denoting factors, we have the value of x equal to the other, we formed a new equation, (a--b)y=ad-h; which contained only the other unknown quantity y. This pro adh cess is called extermination or elimination. Whence, y = In the resolution of equations there are three methods of extermination, viz., by comparison, by substitution, and by ad- --bd Consequently, « = d addition and subtraction. b -6 CASE I.—To exterminate one of the two unknown quantities The rule given above may be generally applied for the exterby comparison. mination of unknown quantities. But there are cases in which RULE.--Find the value of one of the unknown quantities in other methods will be found more expeditious. cach of the equations, and form a new equation by making one of EXAMPLE (4).-Given x=hy, and ax + bx=ya; to find tho these values equal to the other. Find the value of the unknown values of æ and y. quantity in this equation, by the rules formerly given. Then As in the first of these equations « is equal to hy, we may in substitute this value of the one unknown quantity in either of the the second equation substitute this value of x for # itself. Tho other equations, and resolving it by the same rules, the other second equation will then become, ahy + bhy = ya. ven known quantity will be found. The equality of the two sides is not affected by this alters EXAMPLE.-Given x +y = 36, and m y= 12; to find the tion, because we only change one quantity ~ for another which values of t and y. is equal to it. By this means we obtain an equation which viz, 2h a a a y = 4. 2+1 a contains only one unknown quantity. Whenco, y=ah +bh, add them together if the signs are unlike; the result will be an and x = ah + bh'. equation containing only one unknown quantity, which is to be This process is called extermination by substitution. resolved as before. CASE II.-To exterminato an unknown quantity by sub It must be kept in mind that both members of an equation stitution. are always to be increased or diminished alike, in order to preRule.— Find the value of one of the unknown quantities, in one serve their equality. of the equations, in terms of the other unknown; and then in the EXAMPLE (9).-Given 2x + 4y= 20, and 4e + 5y = 28; to other equation SUBSTITUTE this value for the former unknown find the values of æ and y. quantity. From this equation, find the value of this unknown Here, multiplying the first equation by 2, we have, quantity, as before. 4x + 8y = 40. EXAMPLE (5).-Given x + 3y=15, and 4x + 5y=32; to find Subtracting the second equation from this, we have, the values of æ and y. 3y = 12; Hore, transposing 3y in the first equation, we have, Whence, y=4, and a=2. = 15 - 3y. In the solution of the succeeding problems, either of the three Substituting the value of x in the second equation, we have, rules for exterminating unknown quantities may be used at 60 -- 12y + 5y = 32; pleasure. That quantity which is the least involved should be the Whence, by transposition, etc., one chosen to be first exterminated. The student will find it a useful exercise to solve every And, from the first equation, example by each of the several methods, and carefully to observo a = 15 -- 12 = 3. which is the most comprehensive, and the best adapted to There is a third method of exterminating an unknown quantity different classes of problems. from an equation, which, in many cases, is preferable to either added to the namerator, the fraction will be equal to $; but if a EXAMPLE (10).-To find a fraction such that, if a unit be of the preceding. EXAMPLE (6).—Given x + 3y=a, and x-3y=0; to find the unit be added to the denominator, the fraction will be equal to values of & and y. Let x = the numerator, and y=the denominator. Here, if we add together the first members of these two equa Here, by the first condition, we have tions, and also the second members, we shall have, y 2x = a + b, And by the second, we have an equation which contains only the unknown quantity x. The y+1 other, having equal co-efficients with contrary signs, has disap. Whenco, a = 4, the numerator; peared. Still the equality of the sides is preserved, because we And y=15, the denominator. have only added equal quantities to equal quantities. Therefore, it is the required fraction. a + b EXERCISE 38. 1. Given 2c + y = 16, and 3x – 3y = 6; to find the values of x and y. 3 6 2. Given 4x + 3y = 50, and 3x – 3y = 6; to find the values of cand y. EXAMPLE (7).--Given 3:+y= h, and 2c +y=d; to find the 3. Given 3x + y = 38, and 5x + 4y = 68; to find the values of x and ya values of a and y. 4. Given 4x - 40 = ~4y, and 6x - 63 = -7y; to find the values of x Here, if we subtract the second equation from the first, we 5. The numbers of two opposing armies are such, that the sum of shall have a h-d, where y is exterminated, without affecting both is 21,110; and twice the number in the greater army, the equality of the sides. Whence, y = 3d - 2h. added to three times the number in the less, is 52,219. What EXAMPLE (8).--Given -2y = 0, and x +4y=b; to find is the number in each army ? the values of æ and y. .6. The sum of two numbers is 220, and if three times the less be taken from four times the greater, the remainder will be 180. Here, multiplying the first equation by 2, we have, What are the numbers ? 2x - 4y= 20; 7. The mast of a ship consists of two parts; one-third of the lower Then, adding the second and third equations, we have, part added to one-sixth of the upper part, is equal to 28 feet; and five times the lower part, diminished by six times the upper 6 + 2a; part, is equal to 12 feet. What is the height of the mast ? Whence, {=}(b + 2a), 8. What two numbers are those, whose difference is to their sum as 2 And y=3(6-a). to 3; and whose sam is to their product as 3 to 5 ? 9. To find two numbers such that the product of their sum and This process is called extermination by addition and sub difference shall be 5, and the produci of the sum of their traction, squares and the difference of their squares shall be 65. EXERCISE 37. 10. To find two numbers whose sum is 32, and whose product is 210. 1. Given 8x + y = 42, and 2x + 4y = 18; to find the values of u and y. 11. To find two numbers whose sum is 52, and the sum of their 2. Given 2x + 8y=81, and 4+ 6y = 63; to find the values of 2 and y, squares 1,421. 3. Given 3x + 3y = 72, and 4x + 5y = 116; to find the values of r 12. A certain number consists of two digits or figures, the sum of which is 8. If 36 be added to the number, the digits will be 4. Giveu 4x + 10y = 124, and 2x + 9y = 134 ; to find the values of x inverted. What is the number? and 3. 13. The united ages of A and B amonut to a certain number of years, 3. A privateer in chase of a ship 20 miles distant, sails 8 miles, while consisting of two digits, the sum of which is 9. If 27 years the ship sails 7. How far will each sail before the privateer will be subtracted from the amount of their ages, thọ digits will overtake the ship ? be inverted. What is the sum of their ages ? 6. "The ages of two persons, A and B, are such that seven years ago A 14. A merchant having mixed a quantity of brands and gin, found it was three times as old as B; and seven years hence, A will be he had put in 6 gallons more of each, the comiponnd would have twice as old as B. What is the age of each ? contained 7 gallons of brandy for every 6 of gin; but if he had 7. There are two numbers, of which the greater is to the less as 3 to. put in 6 gallons less of each, the proportions would have been 2; and their sum is the sixth part of their product. What are as 6 to 5. How many gallons did he mix of each ? the numbers? CASE III.—To exterminate an unknown quantity by addition KEY TO EXERCISES IN LESSONS IN ALGEBRA. and subtraction. EXERCISE 34. Rule.- Multiply or divide the equations, if necessary, by such 1. ds + 5a'h + 100'l' + 100?h' + 5d1* + ho. factors that the term which contains one of the unknown quantities 2. b* + A*-y + Bb - Sy? + CUP - Syö + D6*-*y* + eto., in which shall be the same in both equations. Then subtract one equalion the co-efficients which are here represented by A, B, C, etc., are n-1 n-1n etc. from the other, if the signs of this unknown quantity are alke, or respectively , and y. 3a aud y. 2 3 |