182 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



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

 integral, except the first, a factor p n cos n<f> or p n sin nfy, so that V P 

 is developed as a function of its coordinates p, </>, in an infinite 

 series of circular harmonics, the coefficients of which are definite 

 integrals around the circumference, involving the peripheral values 

 of V and dV/dn. 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 ra. Then at 

 the circumference we have 





and 



1 n= /n) \ /*2ir 



- 2 n R m - n (- + 1 T m cosn(a>-6)da>. 



\ n /Jo 



l 



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

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

 values of p less than R, the coefficient of every power of p except 

 the mth must vanish, and we have the important equations 



fftr 



(11) I T m cos n (w <) da) = 0, ra =*= n, 



J o 



i f 27r 



(12) T m (<)=- I T m (co)cosm(ct) <j>) da, 



for all values of n, and for all values of m except 0. Since T is a 

 constant, we evidently have 



T =*rT dco. 



Z7T./0 



These two important results can be very simply deduced by direct 

 integration, inserting the value of T m (o>), but we have preferred 

 to deduce them as a consequence of Green's formula (12), 92, in 

 order to show the analogy with Spherical Harmonics. Let us 

 now suppose that the function F(o>) can be developed in the 

 convergent infinite trigonometric series 



00 00 



F(o>) = 2 (A n cos no) + B n sin nw) = 2T n (o>). 







