14 TJ. S. COAST AND GEODETIC SURVEY. 



Let us introduce as a new variable a function of <p 

 denoted by u and defined by the equation 



p dip u d(p 

 But 



Ids^ 1 be 

 p dip sin 6 h(p 

 hence 



1 be _ Idu 

 sin e b(p ud<p' 



By integrating this partial differential equation with respect 

 to (p, we obtain the required relation. This integration may 

 be carried through in the following manner. 



JS^eb-/"'^-} 



du 

 u 





cos^^+sin^TT 



2^"^2d^, Cdu 



sin e b(p 



*=-/? 



„ . e e 



2sm2COS2 





p e r ' ^ 



jsm^ jcos^ 



log sin 2 — log cos 2 = — log t^ + log T (X) .* 



Log r(X) is a function of X that is added since the integration 

 is partial with respect to (p. The function r(X) is as yet 

 undetermined. 



or 



; . ^ 1 r(x) 



log tan2=log-^ 



^ ^ r(x) 



tan o=— ^ 

 2 u 



♦This function has no connection with the gamma function defined by the second 

 Eulerian integial. 



