Motion of a System of Bodies. 49 



tively and we have by (m'), N, - Sm((cf'2 — a"y)P+(a"x — «z)Q-|- 

 (ay-a'x)R,) (0'); by («'), and (g), a'z-a"y = cy'-bz', a"x-az 

 c'y'-b'z', ay-a'x=c"y'-b"z'\ .'.put cP+cQ+c' R = R', &P+ 

 J'Q+rR=Q' also, aP+a'Q-ffl"R=P', (/>'), and we have N, = S»i 

 (R'y' — QV), in the same way (by making some very obvious chan- 

 ges in (0'),)vve haveN // =Sm(P'2 , -R'x'), N„ =Sm(QV— Py'), (q'),. 



Multiply (t") by m, take the finite integrals relative to all the bodies 



r u e ( dx*+ dy*+dz\ 

 of the system, put z>m\ j- J =21 =the living force of 



the system, and we have by (V), (since p, q, r, are the same through- 



/A» 3 +Bo 2 -f-Cr 2 

 out the system ;) T= f 2 1 — Dqr — E/?r — Fpq+ t Cp 



(dx'*+dy' 2 +dz'*\ 



+,Bgf-KAr+Si»f ^ j>( r ')j bv tak,n S the P ar,iaI 



differential co-efficients of (r') relative to p, y, r, we have by (?'), 



dT\ /dT\ „ /«n\ 



$-)^+A (^H'+A (y)-*'+A (O 5 by substitu- 



ting from (*') and (q') in (26), they will be changed to d [-j- I -f- 



1t 

 (dT\ fdT\ _ „. , fdT\ fdT\ fdT\ 



dt 



(dT\ /dT\ IdT 



(P'z-RV),^ j +p{j q ) -q[^J = Sm(QV- Py),(27) : also 



<fr 



substituting from (*') in (25), they will be changed to dy a"{ -p] -f- 



4i> ("> (t, ) )=-■ (->• 



By (e), x'=ax+a'y +a"z, y'=zbx+b'y+b"z, z' = er+c'y-f-e"*, 



(Oj'" -supposing o,6,c,&tc. to be momentarily constant, we have -37 

 adx±a' dy + a" dz fy Jbdx +b' dy + b" dz d£ cdx -f c'dy + c"<fe 



(«'); hence, and by (g'), L=^, M=J-, N=-^, («') 5 substittf- 



Vol. XXVI.-No. I. 7 



