ARISING FROM INEQUALITIES OF TEMPERATURE. 695 



groups we must consider the sextuple integral (20). But we shall not find 



it necessary to do this for terms of not more than three dimensions, for in 



these, if both groups of symbols occur, the index of one of them must be 

 odd, and the integral vanishes. 



We thus find from equations (3), (4), (5), (6), and (7) 



St 



St 



S Ip , 



5-. == - ( - 2a" + a/?- + ay") (33) 



ot 2 a 



St 6 it 



8 



St^r- ~2^ r (35 >- 



[Any rational homogeneous function of 77 is either a solid harmonic, or 

 a solid harmonic multiplied by a positive integral power of (f + ^ + C 2 ), or may 

 be expressed as the sum of a number of terms of these forms. 



If we express any one of these terms as a function of u, v, iv, V and 

 the angular coordinates of V, we can determine the rate of change of each of 

 the spherical harmonics of the angular coordinates. 



If we then transform the expression back to its original form as a function 

 of f ~n\, i> > *? 2 > 4. and if we add the corresponding functions for both 

 molecules, we shall obtain an expression for the rate of change of the original 

 function. 



Thus among the terms of two dimensions we have the five conjugate solid 

 harmonics 



