390 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



virtue of the factor ? therefore if the series 69) converges, the 



-pn f 



series 70) does as well. Since the series fulfils all conditions, by 

 Dirichlet's principle it is the only function satisfying them. 



We may fulfil the outer problem by means of harmonics of 

 negative degree. Taking n negative , the series 



71) 



RT\ 



Q 



is convergent , takes the required values on the circumference, and 

 vanishes at infinity. For a ring-shaped area between two concentric 

 circles, we may satisfy the conditions by a series in both positive 

 and negative harmonics, 



72) 



H- ^? Q~* n {A! n cos no 



140 a. Development in Circular Harmonics. We may use 



the formula 63), 139, to obtain the development of a function in 

 a trigonometric series on the circumference of 

 a circle. Let the polar coordinates of a point 

 on the circumference of the circle be B, o and 

 of a point P within the circumference 0, cp. 

 Then we have for the distance between the 

 two points 



rig. 139. 



Removing the factor R 2 , inserting for cos (CD Q) 

 its value in exponentials, and separating into 

 factors we obtain 



73) 



r = E 



Taking the logarithm we may develop 



and 



