272 Trans. Acad. Sci. of St. Louis. 



Differentiating (5), regarding d, t and or as variable, we 

 have, 



— s\\\d cos< dd — co8^ smt dt = — sine sin^ sina da. 

 By (10) and (6), 



cos^ dt = sine cosq da, 



•% 



or, 



coad da = — sinC cos^' cZa. (H) 



Wethenhave from (8), (10), (9) and (11), 



jd = sin5sincc?a — cosqd^ ) M9\ 



cos^ Aa = — cosq sinC da — sing d^ \ ' ^ 



in which j8 and Aa are the sum of the differentials in 8 and a 

 respectively which depend upon the differentials in a and 

 in C. 



If /<j and ^2 represent the vertical and horizontal flexures 

 respectively, in arc, of a great circle we may, since they are- 

 small quantities, substitute them in eqs. (12) for d^ and 

 8inCc?a respectively. 



We then have to second powers of h^ and A, 



Ad = h^9\nq — h^co^q 

 cos5 Ja = — h^cosq — /ijninq. 



Issued November 21, 1896. 



