351 REPORTS ON THE STATE OF SCIENCE. 



closely connected with Eiemann's formula for fractional differentia- 

 tion, viz., 



C;',.)'" w =r(»^Cte)'"j' (j - 2) "'^*' (2) ' b - 



a 



Abel's integral equation occurs in many investigations, some of 

 which are of a physical character. It is used in the derivation of 

 Schlomilch's expansion [ of an arbitrary function in a series of Bessel's 

 functions of the type 



f(x) = S,aJ (nx). 



A good account of this theory is given in Whittaker's ' Analysis ' 

 and in Nielsen's ' Cylinderfunktionen,' where another proof of Abel's 

 formula is obtained. The proof is intimately connected with an artifice 

 due to Poisson 2 in which an integral over an octant of a spherical 

 surface is expressed in two different forms, e.g., 



TT IT IT TT 



•j 2 S 7 



(/' (sin sin f) sin 6d(kl<j> = [ [/'(cos 0) sin 6 ddd<p 



00 00 



= 2 [/(l) -/(o)]. 



This artifice and its generalisations have been used to obtain proofs 

 of Abel's formula by Tait, 3 Beltrami, 1 and Gwyther. 



A form of Abel's equation has been used by Lamb in the solution of 

 problems in hydrodynamics and the diffraction of light. 



if /H-Jxfr-M* 







CO 



then x(f') = —-- \ /'(/) cosh u) du. 



*- J' 



This formula is a particular case of the more general formula 

 9(V) = [ ;' (,/)f ^ 8f(o) = 



V 



dy 



_ sin Xn d_ f g(y)dy 

 *") - ~ v d v ) In -vf 



which may be deduced from (3). The case in which c = 00 corresponds 



to the case in which a = 0, in (1), and can be examined without difficulty. 



The last equation occurs in the solution of the problem proposed by 



Benndorf and Schuster of determining the velocity of propagation of an 



1 Ztitsehrift filr Math, in Phyxik, 1857. Eecent investigations have been given 

 by Gwyther, Mess.^ Math., vol. xxiii., p. 97, and Miss Smith, Airier. Trans., 1907. 



2 Journ. de I' E cole Poly technique, 1821, Cah. 19. 



3 Proo. Boy. Soo. of Edinburgh, 1874 ; Scientific Papers, vol. i., p. 245. 

 * Rend. Lomb., 1880, ser. 2, vol. xiii. 



