MECHANICS. 



Here a ia repeated three times as a factor in tho multiplicand, 

 and twi.-c in tin- multiplier ; hence, it is repeated Jive times in 

 tho prt'iluc:, :m<l i - cull, d the fifth power of a. 



EXAMPLKH. (1.) What if) tho product of 6666 by 666? Ant. 

 11,1,1,1,1,1,. or /.'. 



What is the product of aa x aaa x aaaa by aaa X aaaa ? 

 AUK. aaaaaaaaaaaaaaaa, or a 1 *. 



7'.i. It is also plain, from Art. 73, that the numeral co-eficienit 



of several factors should be brought together and made into 



tor by multiplication. Thus to multiply 2a X 36 by 



4a X 5'', k'ivcs tho product of 2a X 36 x 4a X 56, or 120aa66; 



For tho co-efficients are factors [Art. 24], and it is immaterial in 



iiese are arranged. Therefore 2a X 36 X 4a X 56 



= ''y3x4x5XttXaX6x6 = 120aa66. 



MPLES. (1.) What is the product of 3x X 4r X 5y by 

 2i/ X 4: y Ans. 48Qxf<j'j~. 



What is the product of 3a X 46h by 5m X Cy ? Ans. 

 '<u/. 



(.'{.) What is tho product of 46 X Gd by 2x + 1 ? Ans. 

 48bcb + 246(1. 



80. The product of two or more powers of the same quantity i.- 

 expresscd by irritiny tlmt quantity with an index equal to the SUM 

 of the indices of the proposed powers. Thus tho product of a s 

 and a* is a* ; and the continual product of x s , x 4 , and x* is a: 1 *. 

 So likewise the product of x m and x n is x m -j- ", and that of x and 

 n is x n + ' ; and, on the same principle, the product of x m n 

 and x n is x m . The reason of this is evident from Art. 79. Thus 

 a 2 and a* are the same as aa and aaa ; the product of which is 

 aaaaa or a* ; tho index 5 being tho sum of the indices 2 and 3, 

 the numbers which show how often a is used as a factor in tho 

 given powers. 



EXAMPLES. (1.) What is the product of a 2 and a* ? Ans. a 7 . 



(2.) Find the continued product of a 2 , ab, anda 4 6* ? Ans. a 7 6 J . 



(3.) Find tho continued product of x 1 , x*t/, z*y s , and xy 4 . 

 Ails, x i/ 



MECHANICS. V. 



PARALLEL FORCES. CENTRE OF GRAVITY. 



BEFORE proceeding to the subject of the Centre of Gravity, 1 

 must direct your attention to two consequences which flow 

 directly from the principles established in the last lesson, anc 

 furnish the basis on which tho properties of that centre rest 

 Yon have seen there that the centre of a system of paralle 

 forces is found by cutting in succession certain lines which join 

 certain points in certain definite proportions, namely, inversely 

 as the forces acting at their extremities. Now, such cutting can 

 give for each line, and therefore for all, as final result, only one 

 point. For example, the centre of two parallel forces of six anc 

 four pounds acting at two points, A B, of a body, as in the las' 

 lesson, is got by dividing A B into ten parts, and counting of 

 four parts next to A, or six to B, and the result evidently can be 

 only one point. If we now suppose a third parallel force of 

 five pounds added, acting at some other point c, of the body, 

 and join the point last found with c, and divide the joining line 

 into fifteen parts, taking ten next to c, here again only one 

 point is the result. And so on for any number of forces it can 

 be shown that there is but one centre. 



But, lent it should be thought possible that, on catting these 

 men in a different order of the point*, ABC, etc., a Moond 

 Centre should turn up, let UH think that possible, and apply 

 orces at these points parallel to each other, but not parallel iu 

 he line joining these two centres. Their resultant then pMSM 

 hrough both of these point*, and therefore mtut act in the line 

 them, which in impottible ; since, as I have proved, it 

 must be parallel to its components. 



Furthermore, you will observe that all these lines are ctit 

 nly in reference to the magnitude* of the forces ; no account u 

 .aken of their direction. Whether they pull upwards or down- 

 wards, or obliquely to left or to right, so long as the magnitudes 

 remain the same, or even keep the same proportion say that of 

 six, four, and five the centre cannot change. Of coarse, tho 

 points are supposed not to change. Whatever be the number 

 of points and forces this is true ; as for three, BO for any otbt'r 

 lumber. And mark, moreover, that it makes no difference how 

 ,his change of direction is produced, whether, leaving the body 

 n one fixed position, you make the forces change directions as at 

 a and 6 (Fig. 17), or, preserving the direction, you turn the body 

 round, as from a to c in the same Fig. In neither case does the 

 centre change. These results may be summed ap in the two 

 allowing propositions : 



1. A System of Parallel Forces acting at given points in a 

 body, has ONE Centre of Parallel Forces, and only one. 



2. The Centre of Parallel Forces does not change its position 

 when the direction of the forces is changed in reference to tho 

 body. 



THE CENTRE OF GRAVITY. 



The centre of gravity is tho particular case of the centre we 

 have been last considering, in which the forces are those by 

 which bodies on the earth's surface are drawn by attraction 

 towards its centre. The smallest body, particle, or atom, is 

 drawn in proportion to its mass, equally with the largest ; and 

 it is to the tendency of these bodies so to move downwards in 

 obedience to this attraction, that we give the name of " weight." 

 The term "gravity" carries a similar meaning, being derived 

 from the Latin gravis, heavy. 



Now, since every particle of matter is thus drawn to the 

 earth's centre, it is evident that the weight of all large masses, 

 such as a block of marble, beam of timber, or girder of iron, 

 ia the joint effect, or the resultant, of the attractions of the sepa- 

 rate atoms. But these attractions are all so many parallel 

 forces ; for, pulling, as they do, towards the earth's centre, 

 which is nearly 4,000 miles away down in the ground, they in- 

 cline, even in the largest objects, so little towards one another 

 that practically they may be considered not to meet, that is, to 

 be parallel. Hence you see that all the principles we have 

 proved about parallel forces hold good of the earth's attraction 

 of these atoms, and that we may aflirm that 



1. A body has one Centre of Gravity, and only one. 



2. The Centre of Gravity is not changed by the body being 

 turned round after any manner in any direction. 



It thus appears that the weights of all the separate atoms of 

 any mass of matter are equal to a single weight supposed to act 

 at some point within that mass, or, as sometimes happens (and 

 we shall see), even without, equal to their sum. There is great 



advantage in this simplification ; for, instead of having to con- 

 sider millions of diminutive forces acting at all its points, 

 we direct our attention to only one force, acting at only one 

 point. 



