87-89] CONJUGATE PROPERTY. 127 



88. Two surface-harmonics S, S are said to be conjugate when 

 ffSS dv = (1), 



where SOT is an element of surface of the unit sphere, and the 

 integration extends over this sphere. 



It may be shewn that any two surface-harmonics, of different 

 orders, which are finite over the unit sphere, are conjugate, and also 

 that the 2n + 1 harmonics of any given order n, of the zonal, tes- 

 seral, and sectorial types specified in Arts. 86, 87 are all mutually 

 conjugate. It will appear, later, that the conjugate property is 

 of great importance in the physical applications of the subject. 



Since SOT = sin 6&6$a) = S//.Sa&amp;gt;, we have, as particular cases of 

 this theorem, 



f P m (M)&amp;lt;^ = (2), 



(3), 



and I T m * (LL) . T n * (/x) dp = (4), 



J -i 



provided m, n are unequal. 



For m n, it may be shewn that 



i 9 



Finally, we may quote the theorem that any arbitrary 

 function of the position of a point on the unit sphere can be 

 expanded in a series of surface-harmonics, obtained by giving n 

 all integral values from to oo , in Art. 87 (6). The formulae (5) 

 and (6) are useful in determining the coefficients in this expansion. 

 For the analytical proof of the theorem we must refer to the 

 special treatises ; the physical grounds for assuming the possibility 

 of this and other similar expansions will appear, incidentally, in 

 connection with various problems. 



89. As a first application of the foregoing theory let us 

 suppose that an arbitrary distribution of impulsive pressure is 

 applied to the surface of a spherical mass of fluid initially at rest. 



* Ferrers, p. 86. 



