78-81 ] General Theory 79 



at all points of the boundary. This equation must be satisfied when e = 0, 

 and for all values of e for which e* may be neglected. Equating coefficients, 

 we obtain the necessary and sufficient conditions of equilibrium, 



JA- "A- -, "*-i '=-,. -(205), 



irpabc %7rpabc a 2 



d\ =0P ............... (208). 



o A 



Equations (205) to (207) are naturally the same as equations (91) to 

 (93) of 52; it follows that 6 is the same as before, and that a, b, c as 

 functions of /u, and &> 2 are also the same as before. 



Since the value of Vi given by equation (202) must satisfy V 2 F i - = 4?r/), 

 we at once have 



o A abc 



giving, on differentiation with respect to e, 



V'f '&,-<), 



Jo A 



so that, in virtue of equation (208), V 2 P = 0, and P is a spherical harmonic. 



80. Not every spherical harmonic will give a possible value for P . 

 For, from the general value of < x as given in equation (203), it is clear 

 that a term in P of degree n in x, y, z will give rise to terms of degrees 

 n, n 2, n 4, ... in $ lt and so to terms of similar degrees on the left-hand 

 of equation (208). From the form of equations (208) and (203) it is readily 

 seen that the most general form which will be possible for P will be a 

 spherical harmonic containing terms of degrees n, n 2, n 4, . . . , these terms 

 only differing from one another by even powers of # 2 , /*, z*. There will be as 

 many values for P as there are independent spherical harmonics, for when 

 the terms of degree n are given, those of degrees n 2, n 4 can always be 

 determined. Thus the values of P correspond exactly to the different 

 spherical harmonics, although not identical with them. 



81. This result can be obtained rather more simply by direct harmonic- 

 analysis. Poincare* has given the requisite analysis for the special rotational 

 problem ; and inasmuch as the form of equations (203) and (208) have nothing 

 to do with special values of o> 2 and /j,, it is equally true for the more general 

 problem now in hand. 



* Acta Math. 1 (1885), p. 259. 



