140, 140 aj DEVELOPMENT IN TRIGONOMETRIC SERIES. 391 



by Taylor's Theorem, obtaining 



74) logr = logE - 45?- (e n *<-y> + e--(-9)) 



= log B -~ e ~ cos (o> - ?). 



1 



This series is convergent if Q < E, and also if Q = E, unless 



03 = (p. 



Inserting this value of logr in 63), differentiation with respect 

 to the normal being according to E, we have 



75) 



Expanding the cosines, we may take out from each term of the 

 integral, except the first, a factor Q n cosncp or p n sinwgp, so that V P 

 is developed as a function of its coordinates p, cp, in an infinite series 

 of circular harmonics, the coefficients of which are definite integrals 

 around the circumference, involving the peripheral values of V 



and * This does not establish the convergence of the series on 



the circumference. Admitting the possibility of the development, we 

 may proceed to find it in a more convenient form. In order to do 

 this let us apply the last equation to a function V m , which is a 

 circular harmonic of degree m. Then at the circumference we have 



T7" ~D m T> m -r>m 1 rrj 



V m = i JL m , -Q^ = -witf JL m , 

 and 



271 



76) F m (P) = ^*- 



The expression on the right is an infinite series in powers of p, 

 while V m (P) is simply Q m T m . As this equality must hold for all 



