428 Mr. J. W. L. Glaisher on a Class of Definite Integrals, 



(page 36) that " all the great problems of astronomical and tei% 

 restrial refraction depend on the integration of the differential 

 equation " 



,p _ coYvdv . 



in which dR denotes an element of the curvature of the path of 



ds 

 the ray (viz. — , in the usual notation, p being the radius of cur- 

 vature), v = (a being the earth's radius, A the angle of 



incidence of the ray at the surface of the earth, and y the radius 

 vector from the centre of the earth to a point of the ray's path), 



v— sin A 



Y the density of the air —e c f while c and &> (which are very 

 small) are independent of Y, and also therefore of v. 



The limits of integration are a and oo for y, and therefore 

 sin A and for v. 



To effect the integration, Kramp expands the right-hand side 

 of (57) in a series proceeding according to ascending powers of 

 co, viz. 



^ R== _ coYvdv _ co 2 Y{l-Y)vdv 3 co 3 Y(l-Y)*vdv 



c(l-i; 2 )* c(l-^)i 2' c(l-v 2 )f • '"' 



and treats each term separately. 



Yvdv 

 To integrate . z\ n +i > P ut ci== ^ ~~ v > axi( ^ we obtain 



k /»QO 



e c C - n +n {2t-cf-)- n -i{l-ct)e- i dt, 



K being written for 1 — sin A (Kramp, p. 118). 



In the problem of refraction we may neglect powers of c and 

 retain only the first term of the integral, so that we are only con- 

 cerned with the integral 



I t-n-ke-^dt ; 



c 



that is, with 



i 



GO 



l- 2n e- t2 dt, ...... (58) 



