524 Mr. 0. Heaviside. [Feb. 2, 



To corroborate the method of getting (65) we may use (60). For 

 this gives 



n+1) , (67) 



again, as in (65). Now if we endeavour to express w z in a similar 

 manner, we find that it will not work. But direct multiplication of 

 the series in the brackets in (61) by the usual expansion of e~ x 

 gives 



^-Lli-f ,\ i 0*-l)(rc-2) (n-l)(n-2)(n-8) 1 



n-ir ^' ja J8 *J 



(68) 



or, which is the same, 



Now the last line we know to express WL Therefore, by (62), 

 we get 



05* , x n ~ l vn+ l 



and this is the result we shall obtain by multiplying the series on. the 

 right of (63) by the usual expansion of c~*. But (70) is only a 

 special form of a more general formula that will appear later. We 

 may use (44) to convert (70) to circular functions. 



A Bessel Function Generalized. 



22. The generalized expansion of e* may be at once applied to 

 ihze other formula. Thus, we know that the solution of 



<* (71) 



