60 Mr. Spottiswoode — Equations of Rotation [1863. 



From these relations it follows tliat 



Co^o-^o'=0, ||odro-Bo^o=0, k . . . (23) 



= dFo^o-Co|^o = 0, J 



which relations may be also verified as follows : — 

 ^o?^o-^odFo=(VG+T„^)(VH+T,|^)-(VA+T„^+V0o)(VF+TJ) 



= V^dr+VTo(G|^ +HG-Adr-F^)+T;VF-V0o(VF+TJ) 

 V{Vdr-T„(SdF +^F)+T,^F-V0oF-Vdr+ST,iF^}; 



Since 



Gl + FB +Cdr=0, 

 Hffi+Bir+FC=0, 



and 



(0 + S)T-V = O, 



or 



0T=V-ST. 



Hence 



^of^o-Vo=VF{T„^-T,^-V0o} 



= VF{T0,(S + O-V0o} 

 =0. 



From these relations it follows that the first denominator, viz. 

 (A, B, C, F, G, 

 • = A^,' + m,' + C^/ + r^mj&o + G^o^o + H^ol^o) 

 = ^„{AEo + B33„ +CCo + 2(Fdro + GaJo + H?IJ 

 = ^„ V { AH B2 + + 2(F2 + + H^) + 3T, + Sa^} 

 =^oV{S2-2^ + 3T,+ S0,} 

 =^oV 30/+4S0, + ^+S^} 

 = ^,V{(S + 0,)(S + 30„) + ^}. 

 Hence, writing (S+ 0o)(S + 30o) + S=Co, we have, finally. 



From this we may obtain the following system : 



6.C„ -f^,C. J, . . . . (^4; 



