Liquid Spheroid and the Genesis of the Moon. 261 

 equations 



\ V^'df = ¥, \ P^V^f = G^, \ Po"V^^=H, (21) 



I/O Jo t^O 



which are connected by the identity 



j_^C-^/_i_ _i_ ^i_\ , 



«oVo Jo Po \aQ+^ W + f Co^ + fr I 



or } (22) 



2/aAco=(hoW + CoW + «oV)F + 2(ao' + V + Co')Gc + 3H. 3 

 By diflFerentiating this identity we can find the integrals of 



the form r FoXa^,' + yfr)-'dylr; 



e. g.^ we have 



2/aoVo= (4 V + 4^0^ - Sao^F + 5G 



+ 



3(co2-«o')(V-«o^)f Po"XV + f)"'^^. . (23) 

 Hence, after reductions, we find 



+ aoVo^E VCZ'o' + 3co^) + 2 V + 2co^ + 6 V^o^] 



+ aoVoH(V + 3co2)-2(V + Co'); • • (24) 

 and 



+ 3Z.,V-«/V + 2^o') + 2aoWH-2co^ . (25) 



7. We shall now suppose that the ellipsoid is a spheroid, or 

 that aQ = bQ, and take ao^ = &o^=Co^(l+/^) ; then we find 



7 BKo ^Ko ^ BLq „ ^^0 T 



° OOo OCo Oao OCq 



= aoW F (10 + 19/^ + 10/* -{-f) + aoW G(14 + 18/^ + 3/') 



+ ao%^H(4+/2)-2co'X2+/'); • • • • (26) 

 and 



=«XF(5 + 8/2 + 3/*) + ao%^a(7+4/2-/*) 



+ 2ao%'H-2co' (27) 



