93, 94] 



NEWTONIAN POTENTIAL FUNCTION. 



181 



94. Development in Circular Harmonics. We may use 



the formula (12), 92, to obtain the develop- 

 ment of a function in a trigonometric series on 

 the circumference of a circle. Let the polar co- 

 ordinates of a point on the circumference of the 

 circle be R, and of a point P within the circum- 

 ference p, <f). Then we have for the distance be- 

 tween the two points 



r = [R z + p 2 - 2Rp cos (o> - <)]*. 



Removing the factor R 2 , inserting for cos (o> p) its value in 

 exponentials, and separating into factors we obtain 



(7) 



Taking the logarithm we may develop 



log (i- 1 



and log (l-^e-* <<>-*) 



by Taylor's Theorem, obtaining 



(8) log r = log R - - 2 - jL <-*) + <r >-<* ) 



QO _n 



= log R ^L t ~ n cos w (a) <(>). 

 This series is convergent if p < R, and also if p = R, unless 



Inserting this value of logr in (12), differentiation with re- 

 spect to the normal being according to R, we have 



(Q\ 



