392 vnl - NEWTONIAN POTENTIAL FUNCTION. 



values of Q less than JR, the coefficient of every power of Q except 

 the m ih must vanish, and we have the important equations 



77) / T m cosn((o (p)do = 0, 

 o 



78) T m (<p) = ~ TlM) cos m (m - <p) 



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

 a constant, we evidently have 27r 



79) T = ~ 







These two important results can be very simply deduced by direct 

 integration, inserting the value of T m (co) ; but we have preferred to 

 deduce them as a consequence of Green's formula 63), 139, in order 

 to show the analogy with Spherical Harmonics. Let us now suppose 

 that the function F(CD) can be developed in the convergent infinite 

 trigonometric series 



oc oo - 



80) F(o>) = V (A n cos n o + B n sin n o) = V T n (o). 







Multiply both sides by cosm (a (p)d& and integrate from to 2 it. 



81) / F(o) cos m (03 (p)dc3=^. I T n ((ai) cos m(a) w}do. 

 *J ^^ *J 



o oo 



Every term on the right vanishes except the m th which is equal to 

 nT m ((p). Accordingly we find for the circular harmonic T m the 

 definite integral Sft 



82) T m (y) = I F(to) cos m (o cp) do. 



o 



For m = 0, we must divide by 2. 

 Writing for T m (jtp) its value 



Am cos mcp + B m sin wqp, 



expanding the cosine in the integral, and writing the two terms 

 separately, we obtain the coefficients 



83) AQ = ~ I F(ra) d&j A m = I F(o) cos mco do, 



o o 



27T 



B m = I ^C 03 ) sinmodco. 



