Motion of a System of Bodies. 285 



given solid: then ^-jcp,^, can be found so as to satisfy the equations 



Sxy'dm=0, Sx'z'dm = 0, Sy'z'dm=0, (p), S being the sign of integra- 



(dz'\ fdz'\ 

 tion relative to the mass of the solid. Let (^1 (-^j denote the 



partial differential coefficients of z', relative to 4., 6, respectively ; 

 then by (e), (0), (A), {k), z'=cx-{-c'y+c"z={x s'm.^+y cos. .^). 



(dz'\ 

 sin. ^+2; COS. ^, x' cos.(p — y' sin. 9=^ cos.-^-y sin.4^== I jt )> x' 



sin.^ 



sm, (pi-y' COS. (p={x shi.-\^i-y cos.^l.) cos.^-t sin.^= ( ^ J, (^) ; 



(dz'\ COS. (p I dz'\ ldz'\ sm. ^ (dz'\ 



hence x'=^s\n.cp[^]+^^[-^^j,y =003.9 [-d^j-^^[^j^ 



sin. 2?) i (dz'Y ((^-YX cos.2<p/dz'\ 



which give a.y =2^^^^ 1^1^^ j sin.M-^^j j+7i^(-^yj. 



W4./'^^" 2 [ dd l^2sm.&\ d^ l^y^- 2 \d^l 



sin. (p[d'z'^\ , , _ • c^ 1 o T 



^. — ~A~jy J (?")• By assuming Sx'z'dm=0, ^y'z'dm=0, the sec- 



fd'Sz'^dmX /d-Sz'"dm\ 



ond and third of (r) give (^ ^^ j=0, ^- — ^ — 1=0, (s), 



which are the conditions requisite to make Sz^^dtn— to a maximum or 

 minimum, supposing ^ and 4- only to vary. By (q), x'^ -{-y''' -{-z'" 

 =3.34.^2 _j_2;2 = L%/.S(a;'2 4-j/'2)£?TO=:SL2£/m-Sz^=(/n, but 

 SL"Jwj = const..'.S(a;'"+y'")(?m = a maximum when Sz'^dm ■= a 

 minimum, and reciprocally; but S [x''^ -\-y"-) dm = the moment of in- 

 ertia relative to the axis of z',. '-the second and third of (p) require 

 (generally,) this moment to be a maximum or minimum. Put 

 Sx~dm=g, Sy^dm=h, Sz^dm=k, Sxydm=g', Sxzdm=^h',Syzdm 

 =k' ; then by (5) ^z'^dm^sxn. ^ ^ {g sin. ^ 4-4-^ cos. ^ 4*+^^' sin. 

 4- cos. 4')+ COS. ^ ^ k-^2 sin.^ cos. ^ {hf s\u.-]j-\-k' cos. 4')? (')• 



By (s), making the partial differential coefficients of Sz"^dm rela- 

 tive to ^ and 4^ separately =0; we have sin. ^ cos. ^ (^ sin.24' + 

 A 003.^4^+2^' sin. 4 C0S.4' - ^)4-(cos.^^— sin.^^).(A'sin.-4-|-A;'' cos.-4) 

 =0, {{g — h) sin.-^cos. 4'+g'(cos.24' — sin.24'))sin.^ + (A'cos. 4- — 

 ^'sin. 4) cos. ^=0, (m) ; substituting the value of cos.^ from the sec- 

 ond of these in the first, [(_§• — A) sin.4' cos. 4'+^''(cos.24' — sin.24')] 

 X [(^ sin. 24^+ A cos.24^+2_§-' sin. 4^ cos. ■\^ — k).{k' sin. -.p — A' cos. 4') 

 ■^{{g-h) sin.4'Cos. •44-g'(cos.24. — sin.24'))-(^'sin.4'+^'cos.4')] 



