OF THE MOTIONS OF A SOLID BODY. 255 



4- or {gvr cos ^-^ + hu cos 2^f. + hu sin ^^—g sin ^^) — ft^ (-^^2 

 cos 24, + ^M cos 24, — Am cos ~v^+ ^r cos 2^|,) -^f{gu cos 24, 

 — ^gM^ cos ^t^ + A COS *^ — hu ^cos 2^J/+2 Am sin cos 4^ — 2g 

 sin cos ^)= -c2 (Am-^) + a2 (Jiu)—¥ (g) + f{~gu + h), 

 whence the whole equation will be] = (gu+h). {hu — gf 



+ I (a2— 6^). M + / (l—u^) I . ^ {hc^—ha^ + fg) u^ g¥ 



-gc^-hfY 



By solving this cubic equation, we may always find a 

 value of Uy such that both cos (p Sx^z'^Dm — sin cp Sy"z"Dm 

 and sin <p Sx"z"Dm + cos 9 Sij"z"Dm may vanish ; conse- 

 quently their squares and the sum of their squares (SarV 

 DmY -\-{Sg'z"Dmy will vanish, and each of these integrals 

 must vanish separately. 



Having found the angles -^ and Q from this computation, 

 we may determine (p by means of the value of S^y'Dm, 

 which may be obtained in terms of the angles 9 and ^J', and 

 of a^, b^, c^y f, g and A, and making this expression vanish, 



we shall have the value of — m, :--— =-J^ang2<p [, since 



cos~^ — sm^ip ^ ^ ^ 



2 sin cos ^izsin 2^, and (1 — 4sin2 cos^^)=z:cos22^z='(l — 4 

 sin ^<p (1— sin'^rt =: 1—4 sin 2,p-j-4 sin ^(p— (1—2 sin^^)^ — 

 (cos 2^— sin2^)2]. 



By these means we may find the angles d, 4^, and (p, such 

 as to make S^Y'^^^^^' Sx'Vdjw^O, and Si/VdwihO. 

 It might indeed be expected that the equation of the third 

 degree would afford three values of u, and three systems of 

 axes; [since in general the equation {x — a).(x — h).{x — c) 

 =0 must vanish when x is equal to a, to & or c;] but we 

 must observe that u is the tangent of the angle ^ formed by 

 X with the intersection of the two first planes, and as there 

 is no condition to distinguish the plane of x' and y" from 



