1863.] of a Solid Body about a Fixed Point. 61 



with similar expressions for /3, /d^, ; y, y^, obtained by writing the 

 suffixes 1 and 2 respectively for 0. By means of these we may write the 

 equations connecting the variables as follow : — 



Lastly, to complete the transformations, the values of p^^ q^y should be 

 determined in terms of ^, q, r. Now 



m+?^A + a^oiFz=(VA+T,^+V0o)(VH + T,|^) 

 + (VH + T„f^XVB + T,B + T0,) 

 + (VG + T„^I^)(VF4-T,S^) 

 = V^{( A + B)H + FG} + T,T,{(^ + + dT^^} + V^H(0, + 0,) 



+ V|^(0oTx + 0.T„) 

 = VXSH + + T J/^l^ + VH) + V^H(0, + 9,) + V?^(0oTx + W 

 =V{V(S + 0o + 0x) + TJ,}H + (V0„T,+ V0,T, + ^TJ, + VO 



=TJ,{[-(S+0o)(S+0.)(S + 0j+V]H + [0,(S+a,)4-0,(S+0x) 

 + S+(S+O(S + 0,)]?|}, 



since 



V=T„(S + e.)=T,(S+e,)=T,(S+9,). 

 Moreover by (11) we have 



(s+0,)(s+0,)(s+0,)=v, 



and consequently the coefficient of H vanishes. And it may be noticed, as 

 a useful formula for verification, that, from the relations last above written, 

 we may at once deduce the following : 



Again, the coefficient of ^ may be thus written : 



(S + 0o+0.)(S + 0o) + (S + 0o+0x)(S + 0,) + S + (S + 0o)(S+0,) . 

 - (S + 0,)(S + 0„)~ (S + 0,)(S + 0,) 

 = ~ (S + 0.)(S + 0o)-(S + 0.)(S + 0,)~(S+0„)(S + 0,) + S 

 =0, 



F 2 



