206 Mr. C. W. Merrifield on the Geometry of 



but 



NX 1/ \ coso— cos«7 1 — sin'>7 Jit tA 



tan Uq +i>) .tan Uq—p) =■ — ~ ^= tt-" — = tau2 t — s j; 



and, similarly, 



tani(,8 + «).tani(/3-«) = tan2g-|). 



Hence 



This determines the mean latitude of the auxiliary circle from 

 the mean latitude of the given circle very simply. To find the 

 modulus, we have 



• {,_ sip /> __ tan j- (g +/)) — tan \ {q —p) sin a . cos <» 



'~sin5~tan|(5'+ja)+tani(g— jj)~ 'v/(sin^/S — sin^a.sinV 



sin 6 . cos o) 



~ a/(1 — sin^^.sin^ft))' 



whence we may obtain 



tan f = cos oj . tan 9. 

 If we reckon rj and /u. from one pole, but q and /3 from the 

 other pole, tanf ^ — ^ ) becomes — tan f ^ + - j. The equation 

 of the mean latitudes, therefore, in its complete form is 



*^"(f ^i)='°'"-*^"(i±2)- 



That for the modulus remains, without ambiguity, 

 tan f = cos &) . tan 6. 



These formulae remain unaltered for the third case, except 

 that w is to be taken so that F(^,&)) = |F(^,t) instead of |F(^,^). 



In order to get the value of w from <^, we have only to equate 

 <pQ and 0, to o) in the elliptic equations 



cos ^Q= cos ^1 cos 02+ sin 0i sin (Jjc^A^q, 



cos 02 = cos (f>Q cos 01 — sin 0q sin 0i A02- 



The first gives 



— : — ~ = tan id> = tan ft)A(^, to) : 

 sm ^ ^ 



the second gives 



. o 1 — COS0 „ A0+COS0 



sm^ o) = -, , . , , cos^ft)= 1^ , A J • 

 1 + A0 ' 1 +A0 



If, as before, we put siuT= sin ^sin 0, wc have A0=cost; 



