62 C. V. L. Chailier 



WicKSBLL bas given a general symbolical expression for the ß-coefficients of 

 any arbitrary order, which, however, is not reproduced here, as in this memoir I 

 do not carry the development farther than to the second order. 



Substituting the series for f^f,, in (127) and integrating regarding U^, V^, 

 we obtain 



(137) v,oo = ^^^ooü<V.A./]/"(-2f7,^+ rV+ W,')I%,,,,j,,,dU,dV,dW,. 



The three-fold integral has for every a certain numerical value, which 



we denote by so that 



(138) a.j,^ = fJI (—2U-'+ V + W) I <^^j,^dU dV dW 

 and then is 



(139) , V200 = 71^ ^000 yfc "ijk ■ 



45. Computation of a^i- 



We demonstrate the following propositions : 



1) the coefficient Og^^ vanishes; 



2) all ciiji; of odd order (= for which ^ -j- ; -|- = odd number) vanish. 



Let us first suppose that / is a constant as is the case accepting the repulsion 

 law of Maxwell. Then we have (indices are omitted): 



(140) a,.^, = /^(— 2 + F'^ + W) d U d V d W. 



CO 



We now use a theorem from the general theory of the ^.-series in § 34. Let 

 F be any function developed in an /1-series having the generating function <i>, so that 



(141) i^ = 2'ft^,a>^,, 



so is 



(1 42) b^, = III B^., FdUdVd W, 



where Rijk are the polynômes of Hermite. 



Putting F = f^^ijk , we derive that the expression 



R FdUdVdW 



has the value 1, if 



i' = i, j' =j, k' = h, 

 but the value zero if not all these equalities are satisfied. 



