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



Differentiating (5), regarding o, I and a as variable, we 

 have, 



— sind cost dd — cos<? s'mt dt = — sin£ sin<£ sina da. 



By (10) and (6), 



cosd dl = sin:: cos*? da, 

 or, 



coad da = — sin? cosq da. (H) 



We then have from (8), (10), (9) and (11), 



jd = sinq sin J da — cosqdZ > M2\ 



cos<5 Aa = — cosgsinCcZa — s'mq d* )' * ' 



in which Ao and Aa are the sum of the differentials in d and <z 

 respectively which depend upon the differentials in a and 

 in c. 



If \ and h 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 dZ and 

 sm'da respectively. 



We then have to second powers of h 1 and A 2 



Ad = h 2 s'\nq — h^osq 

 coso Aa = — h 2 co&q — filing. 



Issued November 21, 1896. 



