Moment of Inertia. $3 1 



we have, 



v : v' : : R : R', and v : v" : : R : R" (i). 



Letting fall from F the perpendiculars , a', a", upon the direc- 63 * 

 tions of the velocities w, u\ w", we obtain, 



M u a 4- M' u' a' M" u" a" = 0. 



Now if we let fall also the perpendiculars c, c', c", upon the direc- 

 tions z0, w', w", we shall have by article 62, 



M ' U a -f- M'V'R = M * ZO * C, 



or, 



M W a = M W * C -- M * V ' R J 



In like manner, 



M ' u' a' - M' w' c/ M' i/ R'. 

 M" M'' a" = M" w" c" 4- M" i>" R" 



If from the sum of the two first of these three equations we sub- 

 tract the last, we shall have 



M u a 4- M' it' a' M" u f ' a", 

 or 0, equal to the expression below, thus, 



1 = M w c 4* M' w c M" w" c" 



M ' V * R M' ' V ' R' M" ' v" * R". 



f 



But the above proportions f i) give v' = ^- and ?" = ^ 

 substituting these values for v' and i>", the equation becomes 

 = M w c + M' zi/ c' *u." ID" c" 



= M zt> c -f- M' w c M'' w'' c" 

 (M R 2 -f M' R /S -f M" R") 



whence, 



M * w c 4- M ; w cf M" -w" c" 



M R 3 4- uf R /a -I- M' ' ' R ' a 



