342 ON THE iMOTION OF 



Much use will also be made of the subjoined equations : 



£ = «£/+*>, + <•£ l = at+.a!% + a'%<' 



r = a %+b%+c% (8) n, = bZ + b'S'+b'r 

 c" = a %^rb\-^c% I = cH-c'l'4-cr 



dx, = adx -\- a! dx 1 -\- a" dx" d*x^ = ad'x-^-a'd'x' -\-a'iVx 



dy, = Ma: + 6Vta' + fc"rfa;" (9) flty, = /«ra: + b'd'x' + b"d'x" 

 dz, == c<te -(- c'da:' -|- c"(te" <fz a = cd*x -+- c'd'x' -\- c'iVx 



dl = adg-i-a'dZ'-ha"d£" <T& == ad?£ + a!d*% -\-a"d*£ 



dy, = bd£-hb'd£-^b"d£" (10) rf> Q = bd'jt + b'd'g +b"d*Z 

 dt, t = cdg-+-e!d£-t-cW d% = cd*£ + c'rfT -+- c"rf"g 



with similar expressions for the incomplete variations dx,, dy„ <5z„ 

 #£, &?„ d^. Finally we have the following relations between 

 the accelerative forces: 



X = aX,-^b Y.+c Z, X, == aX-ha'X'-ha"X" 



X I = a , X l -hb'Y,-hc'Z l (11) Y, = bX-t-b'X + b"X" 

 X" = a"X,-Y-b"Y,+c"Z, . Z = cX -+- c'X -+- c"X" 



If we substitute now, in place of the variations and accele- 

 rations of x, x', x" in the above formulas, their values derived 

 from equations (2), and reduce by means of (6) and (10), we 

 shall find 



(12) 



dx, = d£, — y,dR-hz,dQ 



by x = d*i,— z,dP -\- x,8R 

 dz, == dZ — x.dQ + y.dP 



(13) 



dX = d% — x,(d& -h dR) + y,(dVdQ — d*R) -f- z,(dRdP -+- </ J Q) 

 ,/'y a = dX — #(</#' -+- dPJ + z,(rfQrf« — d"P) -+- x,(dPdQ + rf a B) 

 ,Pz a = d^—z l (dP'+dQ*)-hx,(dlidl>--d*Q) + y l (dQdR + d*l') 



