M KC HANK'S 



EXAM PLES. Find the squares of -f- i s and of a 

 II. -re, * + 4 Also, | 



.-4 



Otherwise, reducing the mixed quantities to improper frac- 

 wo have a + $ = ^ * ; and - 4 = ?il_?. Whence, 



, and r* a> 



^ (2* - U = 



2 ' 

 , aa before. 



EXERCISE 34. 



. 



1. What is the 5th power of (d + h) t 

 U. What is the nth power of (6 + y) ? 

 :i. What is the 6th power of (3x + 2y) t 



4. What is the 2nd power of (a b) ? 



5. What is the 3rd power of (a I)? 



6. What is the 4th power of (o b) ? 



7. What is the Cth power of (* y) f 



8. What is the nth power of (a b) t 



9. What is the 4th power of (a 1) t 



10. What is the Cth power of (1 y) ? 



11. What is the nth power of (I + ) P 



o 



12. Find the square of a + _. 



p 



Find the square of x -. 



I 



13. 



14. Find the square of + Sxy. 

 m 



15. 



Find the square of - + 2abc. 



EXERCISE 35. 



1. Expand (x + i;)'. 



2. Expand (a + b)*. 



3. Expand (a b)*. 



4. Expand (x + y) 5 . 



5. Expand (x y)*. 



6. Expand (m + n) 7 . 



7. Expand (a + b)'. 



8. Expand (* + i/) 10 . 



9. Expand (* y) 1 *. 



10. Expand (o b)'. 



11. Expand (a + b)V 



12. Expand (2 + *)*. 



13. Expand (a b* + c). 



14. Expand (a + 3bc). 



15. Expand (2ob i)'. 



16. Expand (4ob + 5e*)' 



17. Expand (3* 6y)*. 



18. Expand (5a + 3d)*. 



KEY TO EXEECISES IN LESSONS IN ALGEBRA. 

 EXERCISE 31. 



7. 64b. 



1 1 1 



8l ' r " 



9. 216m*y. 



10. a"b. 



11. 64a e *. 



12. 1296a 1 * A. 



EXERCISE 32. 

 , a* (d + TO)* 

 (*+!) ' 



5. ' 2ab + b. 



6. o 3 + 3a + 3o + 1. 



7. a" + 2<zb + 2aJi + b 



+ 2b7i + 7i. 



8. a + 4ad + 6a + 4<1 



+ 12d + 9. 



EXERCISE 33. 



13. (o + b) 10 . 



14. (a + b)"n. 



15. (x y)". 



16. (* + !'). 



17. a'b'. 



18. aVh l> . 



9 b 

 10. 



+ 8b + 24b + 

 32b-H6. 

 > + Cx* + 1(U + 

 ID* 1 + 5* + 1. 

 11. 1 Cb + 15b 

 2Cb + 15b* 6b 

 + b. 



1. 4a + b + 4ab. 



2. h* + 1 + 27i. 



3. a*b + c*d + 2abed. 



4. 36y 



5. 9d + h 6dJi. 



6. o + 1 2a. 



MECHANICS. XXIV. 



IMPACT CENTEIFUGAL FOECE THE PENDULUM- 

 CENTEE OF OSCILLATION. 



WHEN two bodies striko one another, they touch first in some 

 one or more points, and the motion of these is usually com- 

 municated to the whole body. Thus, when a carpenter strikes 



.,! I,,- ,....:, h* 

 by the whole. 



i a nail with his hammer, it only timsiin 



* i . ,. , t i ., ti - - ^ i. 



w6 EDoniemMuD ftoqawQ D/ uui ptn w 



If, however, the blow b* not true, the head alone' may receive 



the motion, and fly off by tUeH , leaving the rest unmoved. 



Sometimes, especially if the body .truck be soft or brittle, 

 and the velocity of the other be great, there is no time for the 

 motion to be thus shared, and then the shape of the mass ir 

 altered, or the part strode flies off M a chip. A homely illus. 

 tration of this is afforded by a simple experiment whioh all 

 may try. 



Balance a small piece of card on one of the fingers of the left 

 hand, and lay a shilling on the top of it By a sodden blow 

 with the finger and thumb of the other hand the card may be 

 jerked away without moving the shilling. Care must, however. 

 be taken to strike the card exactly in the direction of it* surface, 

 as if it be tilted up or down the shilling will, of course, be jerked 

 off. After a few trials, however, you may be pretty certain of 

 success. The explanation is, that the motion of the card is so 

 rapid that it has moved quite away before it has had time to 

 communicate its motion to the shilling. 



There are many other familiar examples of this, some of which 

 verge on the marvellous. 



If a bullet be fired at door net half open, it will pass through 

 the panel without shutting the door or moving it on its hinges. 

 We may even go further, and, instead of a bullet, put a tallow 

 candle into the gun and fire it at the door, it will be found to 

 pass through instead of being smashed against it, as we should 

 naturally expect. The velocity of the particles of tallow is so 

 great that they have passed through the door before they have 

 time to alter their relative position. So, if we fire a ball at a 

 window, it will pass through the pane without cracking it, 

 merely making a clean round hole. If, however, the bullet be 

 nearly spent, or its velocity be not sufficiently great, the glass 

 will be shivered to pieces. 



This, too, explains why a good skater will glide swiftly over 

 ice far too thin to sustain his weight. His motion is so rapid, 

 that before the ice has time to yield he has paseed on to another 

 portion of it. We see, then, that a certain amount of time is 

 required for any motion to be imparted from one body to rmr<hfr_ 



CENTRIFUGAL FORCE. 



If a lump of metal or other heavy substance be fastened to a 

 piece of string, and then swung round and round, we shall find 

 that the string is stretched with a strain which varies in pro- 

 portion to the speed with which the body revolves. This strain 

 is called centrifugal force, and is 

 merely one of the results of the first 

 law of motion. 



Let B (Fig. 104) represent a body re- 

 volving round a centre A, and confined 

 by the string AB; its tendency at every 

 instant is to continue in the -n-rr line 

 in whioh it is travelling at that instant, 

 that is, to fly off at a tangent, as 

 along B c. We can easily prove that 

 this is the case, for if, when whirling 

 a sling, we suddenly out the cord or 

 leave the end free, the stone will fly off in a straight line. 



Suppose D to be the point which the stone has reached at 

 the end of one second, then B D will represent the space passed 

 over, and therefore the velocity of B. This we can resolve into 

 two parts, BF acting along the tangent BC, and BO acting 

 along the direction of the cord. The former lepieseiits the 

 velocity the stone has acquired, the latter is the force exerted 

 by tho string to keep it moving in a circle ; this, therefore, re- 

 presents the centrifugal force. We can thus easily see that 

 the greater the velocity with whioh B moves, the greater will 

 be the strain on the cord. If, for example, the velocity be so 

 much increased that at the end of one second B is at E instead 

 of D, the tension of the cord will be represented by B H instead 

 of B a. As, however, this tension always acts in a direction at 

 right angles to the motion of the body, no velocity is destroyed, 

 the only alteration being in its direction. 



We constantly meet with illustrations of the action of this 

 force. A can filled with water may be swung round the head 

 without a drop being spilt When the can is at its highest 

 point, and therefore mouth downwards, the water is attracted 

 towards the earth ; but thin attraction U more than overcome 



Fig. 104. 



