fundamental principle ©/"Laplace's Functions. 505 



which the angles A'OQ = 0, AO P = #' are measured (//,= cos#, 

 pj= cosd')j eAE the great circle from which the angles E AQ = &>, 

 E A P = o)' are measured : p= cos Q P. 



P" *s 



Now an element of the surface of the sphere at P 



I shall change the origin of the angles from A to Q, and make 

 QP = cos -1 /? aDd A QP = i|r the coordinates to P from Q. Hence, 

 dividing the surface of the sphere into elements about Q as the 

 new origin of angles, the element at ~P = — dpd'ty i and the func- 

 tion to be integrated is 



il r-^i-c^(f,' } co')dpdf 



-Jo (l+c 2 -2c# 



the values of p! and &>' in terms of p andi/r should be substituted 

 in F(/a', a)') : this is not done because it will not be wanted. 



I shall first take the case in which F(///, co 1 ) is constant and 

 = 1 : then the integral is 



(l-c^dpd^ 



I 



n 



J - 1 J 



2tt 



J 



(l+c 2 -2cp)* 



Take the point D in Q so that OB = c, then DQ=l-e, 

 Dq = l+c. Join D P and P, take p Q near to P. Draw P M, 

 p m perpendicular and P?i parallel to Qq,pr perpendicular to D P. 

 Then, because Ynp and Vrpavc right angles, a circle can be drawn 

 through P, p, n, r. Hence 



