84 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



which makes the bracketed factors in the two equations 

 equal. Presupposing that 5, a-j, and x. are small quantities 

 we differentiate equations {5), and eliminating a-j, and X2 

 find, when quantities of the order cjr are neglected, 



3 = 



(1 — sin S.^) c 



cos do — sinS.i cot 6^ cos (r^ — ni) 

 If for So we substitute the polar distance, Pi = ^0" — d.^, 

 this equation becomes, very approximately, 



3 = c . tan ipa j 1 -|- cot 8^ tan d., cos (tj — m)l (6) 



Dividing the first of equations (J) by the second, we 

 obtain : 



tan \J (r, + r,) - m - S J = ^.^^ ^^j _ ^-^ tan i (r, - r,) (7) 



We now assume the auxiliary quantities, 



2r = (a, - S') - (a-, - S) 

 U = a, - S - ^T - m - Sr (§) 



and introducing them into (7) find 



tan {r -\- V) = .^/ ^ .- tan r 



sin (6 J — 60) 



whose solution is 



tan U = er>^ 6\ tan S, sin 2r 



1 — cot d'l to7i So COS 2r 



In equations (8) jT+ m is now the only unknown quantity, 

 and to determine m we apply (i) to the polar star and sub- 

 stitute in it the value of tan n given by {2) and the value 

 of Tj — m given by (^) and [S), and find 



sin m = — cot S^ tan (p sin (2r -f C7 + 5) + sin b sec (p -\- sin c tail tp 

 in which terms of the order en- are neglected. Subtract- 

 ing from each member of the equation the auxiliary quantity 

 sin m' = — cot S^ tan (p sin (2r -f- U) {10) 



we obtain to the same degree of approximation 



7n = in' -{- b sec cp '\- c tan cp — ^ cot d\ tan (p cos (2r + U) 



Substituting for ^ its value in terms of c, and introduc- 

 ing into {8) the resulting value of m, we obtain 



/IT+Cc = ao - (S+ U+ m' -f b sec cp) (11) 



