I 



46 Motion of a System of Bodies. 



parallel to itself. If the system is not subjected to the actions of any 

 foreign forces, (11) will exist relative to the moveable origin, whether 

 the changes of the motions of the bodies are finite in an instant or 

 not, and A, ,A, „A, will each be invariable during the motion of the 

 system ; .\ (16) will each be invariable, hence the invariable plane 

 always passes through the moveable origin and moves parallel to it- 

 self. It has been proved when the system is not subjected to the ac- 

 tions of any foreign forces, that the eentre of gravity is either at rest 

 or moves uniformly forward in a straight line ; .*. by fixing the ori- 

 gin at the centre of gravity, the invariable plane is either at rest or 

 moves parallel to itself. Let the origin be at the centre of gravity, 

 and suppose that the system is not subjected to the actions of any 



r • r ii /..n^ fxdy—ydx\ g _ [%dz-zd% 

 foreign forces, then by (11) Sm( -, j =A, Sro( -j- — 



A, (20). Now we have m'((x'-x)dy' 



// 



{if - y)dx')=m'(x'dy' - y'dx')+m'(ydx' - xdy'),m"{{x"-x)dy"- (y" 



-y)dz")=m"(z"dy" -y"dx")+m"(ydx" -xdy"), and so on for all 



the bodies except m : hence Sm'( (x' — x) dy' — (y' — y) dx') 



Sm'(x'dy' - y'dx' ) -f Sto' ( ydx' - xdy'), but Sm'ydx' = ySm'dx', 



Sm'xdy'=xSmdy', and by the nature of the centre of gravity Sm'dx' 



t c T , ( (x' - x)dy' — {y' - y)dx' 

 - max, Sm dy' = - mdy; hencemSm'l- ' — j t 



/(x—x')dy — (y — y')dx\ 

 toA, in the same way m'Smi- dt — )=m'A,and 



so 



/ (#' - x) dy' - (y'~ y)dx'\ 

 on for all the bodies; .'• mSm'(~ LJ fo- ~ ~ ) + 



m'Smf ^* *' y J y y)dx ) -f &c. = to A + to'A -f &c. or 



Smm 



^psa^d^e=msm) = ASm , (21) , i0 the 



i(x'-x).{dz'-dz)-{z , -z)Adx , -dx) x 

 same way Smm'l ^ J = ^Sm, 



((y'-y).(dz'-dz) - (z' - z) Ady' ~ dy)\ 

 Smm' [ Ky sn jL ^~ M J 9 A =„ASm, (22). 



Let the plane x, y be the invariable plane, then A is a maximum, 

 and ,A, „A are each =0 ; ♦'. the first member of (21) is a maximum, 

 and those of (22) are each =0. Let a plane be drawn through any 

 body of the system parallel to the plane z, y, then the first member 

 of (21) is a maximum relative to the parallel plane ; for its value is 



