106 Mr. O. Heaviside. [June 15, 



9 



(2) 



sr \ 6" 3*5- / TT \x x 



1 2 3 2 5 2 





Here A is the usual form of the Fourier-Bessel function (or, rather, 

 the function I (sc) instead of the oscillating function 3 Q (x) t whose 

 theory is less easy), or the first solution in rising powers of # 2 of the 

 differential equation 



as in (71), (72), Part I. Also, B is a particular case, viz., (78), Part I, 

 of the generalization of the same series, (77), Part I, using the odd 

 powers of x, and going both ways, in order to complete the series. 

 And C is an equivalent form of the same function in a descending 

 series, (31), Part I, obtained analytically, before the subject of gene- 

 ralized differentiation was introduced. The analytical transformation 

 from A to C was considered in 14. The present question is, what 

 relation does C bear to A and B algebraically ? It cannot be alge- 

 braically identical with either of them alone, on account of the radical 

 in C. We may, however, eliminate the radical by employing the 

 particular case of the generalized exponential that will introduce the 

 radical anew. Thus, (63), (64), Part I, 



If we use this in (3), and carry out the multiplications, we obtain 

 a series in integral powers of x, positive and negative ; thus, 



^ 



'' 



+(4}! + (iJ)!, U 



I K ~ rl 7 T^ 9i^. I Q ~ . . I | . 



1 f!V (L 3^2 \ -I 



1 . Uy 1.1.2 -^ I \. /g) 



Q ^F W "/ "-!' 



28. Now B involves all the odd powers of #, whilst A involves only 

 the even positive powers. But the terms involving even negative 



