260 CELESTIAL MECHANICS. 1. VU. 28. 



every line of its surface in succession as an axis ; but in 

 more complicated cases it is not so obvious, without a de- 

 monstration, that the whole of some one line must neces- 

 sarily be at rest at each instant. Now] the quantities p, 

 q, r, which have been introduced into the equations (C) 

 (345), are remarkable for affording the situation of the true 

 momentary axis of rotation with regard to the principal 

 axes. For if we take the fluxions of the values of a:', y\ 

 and 2f (345 or 324) and make them vanish, and afterwards 

 take also ^^ziO, which is always allowable, since the posi- 

 tion of the fixed ordinates is wholly arbitrary, we shall 

 have, [retaining the notation of article 345, Ax'r=.dx" ■\-^\/* 



+ yV', and d/=:8V' + sy'+ K't!', or d/= a:"5--dfl. sin 5 



sin "^ sin ^4-d4'.(cos 5 cos -^ sin ^ — sin -^ cos ^) + d^.(cos 5 



sin r|/ cos ip— cos ^ sin (p) > -Vy" \ — d5 . sin 5 sin 4^ cos (p-\- 



d^* . (cos 5 cos -^ cos (p + sin nj' sin ^) — d^ . (cos 5 sin -^ sin (p 



-f cos ^ cos ^) J +J2;"(d5 . cos 6 sin Tj/ + d4' . sin 5 cos 4') ; or 



putting i|/zz0,] dx'=x"(dTj' . cos 5 sin (p — d^ . sin (p)-\-y\A-\', 

 cos fi cos (p — d^. cos ^, -f 2''(d4' . sin 5)— 0. In the same 

 manner we obtain dy'=: 

 3^\ — d5 . sin 5 sin (p — d^' . cos (p -f d^ . cos 9 cos f ) -\-y'\ — dfi . 



sin d coF. ^ -I- d^- . sin (p — d^ . cos 5 sin ^)+jr"d5 , 



cos 5=:0; and dz'iz 

 y( — d5 . cos d sin (p—^(p . sin 5 cos 9) + ?/"( — d5 . cos 5 cos <p 



H-d^ . sin fisin ^)+2"( — dd. sin ^)=:0. 

 If we multiply the first of these equations by — sin ^, the 

 second by cos 5 cos (p, and the third by — sin 5 cos ^, and 

 then add them together, we obtain 



\x" ) — d5.(sin cos S sin cos ^— sin cos ^ sin cos ^) 



