84 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



which makes the bracketed factors in the two equations 

 equal. Presupposing that $, x ly and x 2 are small quantities 

 we differentiate equations (5), and eliminating x lt and x 2 

 find, when quantities of the order en 2 are neglected, 



^ = __ (1 - sin d 2 ) c _ 

 cosd 2 sin 6 2 cotS 1 cos (r l m) 



If for $ 2 we substitute the polar distance, P 3 = 90-S 2 , 

 this equation becomes, very approximately, 



3- = c . tan \p z \ I + cot d 1 tan S 2 cos (T m) I (6) 



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

 obtain : 



' tan * (,, + r.) --- + * i (r, - r.) (7) 



We now assume the auxiliary quantities, 



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

 U ' = a 2 - S - AT - m - 5 (8) 



and introducing them into (7) find 



, , ,.,, sin ($.. + 2 ) . 

 ton(r+ t 7)=- i||(a ' i + a ->tor 



whose solution is 



tan U = cot 6 ^ tanS, tin 2r 



1 cot d l tan 6 2 cos 2r 



In equations () dT-\- m is now the only unknown quantity, 

 and to determine m we apply (1) to the polar star and sub- 

 stitute in it the value of tann given by (2) and the value 

 of T^ m given by (4) and (8), and find 



sin m = cot di tan g> sin (2r -f- U + $) 4~ sin b sec (p + sin c tan <p 

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

 ing from each member of the equation the auxiliary quantity 

 sin m' = cot d^ tan cp sin (2r -f- U) (10), 



we obtain to the same .degree of approximation 



m = m' -\- b sec (p -f- c tan cp 5 cot d^ tan <p cos (2r + U) 



Substituting for 3- its value in terms of c, and introduc- 

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



4T+Cc = a a - (S+U+m' + b sec <p} (11) 



