1863.] of a Solid Body about a Fixed Point, 57 



with similar expressions for the two other equations. Multipljring the 

 system so formed by y, y^ respectively and adding, the coefficients of 

 'p\, q\ will vanish, and that of r\ will =1 in virtue of (12) ; and as regards 

 the right-hand side of the equation, the coefficient oi 



Ha+Bai+Fa^, y^ 

 Ga+Fa, + Caa, a„ y^, 

 which, omitting common factors, 



(s+0)A+fi+(s+0)0, VA+ra + v^, 



(S + 0)H+1^, VH+m 



{(s+0)a 



VG + m VG + T,6^ 



VA+T,^ + V02 

 VH + T,?| 



VG+T,^ 

 VH+T,?|(S+0)H+?^ 

 VG + T,ai^(S + e)G+^ 

 (S+0)H+^ VH+ra 

 (S+0)G + ^ VG+TO 



= {(S + 0)0 V(T, - T) + V0( V ~ T,(S + 0)) + V0,(T(S + 0) -- V) } (H^K - JG ) 

 =V(0,-0){T(S+)0-V}(H^-|^G). 



But 



T(S + 0)-V=(S+0)(02 + S0 + ^)-V = (S4-0){(S + 0y-(S + 0) + ^}-V 



= (S + 0)3-S(S + 0y+^(S+0)--V 

 =0. 



Hence, finally, the coefficient of^^* vanishes. 

 So likewise the coefficient of 



A/3 + H/3, + G/3, /3 y =0. 

 H/3 + BAH-F/3, ft y, 

 G/B+Fft + Cft ft y 



And that of r^y 



y 



72 



Ay + Hy, + Gy, y v =0. 



Hy4-By, + Fy, y, 

 Gy + Fy, + Cy2 y^ 

 Similarly the coefficients of and will be found to vanish ; and 

 lastly, the coefficient ofp^ q^ 



= a { A (fty,-fty,) + H(fty -/5y J + G(/3y, -fty)} 

 + a,{H(/3,y,-fty,) + B (fty-/3y,) + F (/3y -fty)} 

 + a.{G(ft7.-/3.7:) + F(ft7-/37.) + C(/3y -fty)} 

 -/3 {A(y,a,— y,aJ + H(y,a-ya,) + G(ya,-y,a)} 

 -ft {H(y,a, - y,a,) + B (y,a- ya J + F (ya ^ - y,a)} 

 ' (y,a, ~ y,a,) + F (y,a- ya,) + C (ya,- y,a)}. 



