58 Mr. Spottiswoode — Equations of Rotation [1863. 



which, by reference to (9), may be transformed into 



□ { ( Aa + Ha, + Ga, )2 4- (Ha + Ba, + Fa,)H (Ga + F«, + Ca,)' 

 -(A/3 + HA + G/3,y+(H/3 + Bft + F/3JH(G/3 + Fft + C/3J^} 



= □ {(AaHBa;HCa/-f 2Fa,.'f, + 2Ga,a + 2HaaJS 

 -(A,(3H B/3,^+ + 2FA/3, + 2Gft/3+ 2H/3A)S 

 + (^-^)(^^-/30 + (i3-^)CV-/3,^) + (C-^)(«,^-/3,') 

 + 2dr(a,a, - A/3,) + 2ffi(a,a - /3,/3) + 2^{aci, -PflJ } ; 



in which the coefficient of S vanishes in virtue of (12) ; so that the coeffi- 

 cient of p^, 



= □ {(^-^, 5S-^, C-^, dT, 05, ?^X^, a,y 

 but, by (12), 



(^-^, B-^, C-^, dT, 05, ?§laaxa,)=^=0, 

 (^-^, 23-^, C-^, dF, ^, P?I/3/3A)'=0,. 



Hence the coefficient in question 



= □(0-0,). (18) 



and the equations of motion become 



9^ = ^(e,-6)r,p„ I ..... . (19) 



^/-□(a-0,)M., 



To find the value of □ in terms of A, B, C, F, G,H, we have from (12) 



Aa + Ha, + Ga,= □ -KPhV-p.yrl 

 A/3 + HA + G/3,= □ -^(na,-y,aj, 

 Ay + Hy, + Gy,= □ -'(^A-^Al 

 Ga + B a, + Fa,= □ -^(fty ~/3 y,), 

 H/3 + B /3, + F A = □ - i(y,a - y a,), 

 Hy + By, + Fy,= n-^(a,/3 -aft), 



Ha+ Fa, + Ca= Q "Kl^ rz-fty), 

 G/3 + FA + Cft=: □'-i(ya,-y,a), 

 Gy + Fy, + Cy,= Q -^(aA-«,^). 

 And forming the determinant of each side of this system, there results 



or 



V (20) 



