356 REPORTS ON THE STATE OF SCIENCE. 



parameter r for which the equation possesses a solution satisfying the 

 boundary condition of the problem. 



The equation (2) was also used by Caque ' in his researches on 

 differential equations, and he expressed the solution in the concise form 



t 



<j,(t) = g(t) + \[K(t,0)g(d)dt) 







where K(t, d) is the so-called solving function. 



The general study of the equation was, however, taken up by 

 Volterra 2 in 96), and the theory was at the same time applied to the 

 equation (18 



f(t)= L(t,8) 9 (e)dd (3) 



which had been previously studied by him, 3 and treated as the limit of 

 a system of linear equations. 



The last equation is named after Volterra in honour of his extensive 

 researches on the subject. 



The general method of treating the integral equations (2) and (3) 

 was also discussed in 1895 by Le Eoux in connection with his researches 

 on partial differential equations. Abel's integral equation, which is a 

 particular case of equation (3), had been extended by Sonine 4 in 1884. 

 Volterra obtained a more general result, and connected it with his 

 general theory. He also considered equations in which there are two 

 variable limits in the integral. 5 



These equations and analogous ones have been treated subsequently 

 by Holmgren, 6 Lalesco, 7 Picard, 8 and other writers. The connection with 

 differential equations has been developed by Fuchs, 9 Dini, 10 Lalesco and 

 the author. 11 



The general formula for the solution of the integral equation 



X 



fix) = *(*) - \Ur,t)m<it 



o 



X 



<t'(x)=f{x)+\k(x,t)f(l)dt, 



1 Liouville's Journal, 1864, vol. ii., t. 0, p. 185. 



2 Torino Atti, 1896, pp. 311, 400, 557, 693 ; Annali di Matematiea, 1897. 



3 II Nuovo Chnento, 1884, t. 4, p. 49 ; Rend. Lined, 1884, ser. 3*. t. 8. 



4 Acta Mathematici, t. 4. Another generalisation was given by P. G. Tait, 

 Scientific Papers, vol. i., p. 245 ; Edin. Pruc. Roy. Soc, 1874. 



5 Equations of this type have been also considered by Picard, Comptes rendus, 

 1909 . 



6 Torino Atti, 1900, t. 35 ; Upsala Memoirs, 1900, t. 3. 

 ' Liouville's Journal, 1908. 



8 Comptes rendus, 1904, p. 245 ; ibid., 1909. 



,J Annali di Matematiea, ser. (2), vol. iv., pp. 36-49 (1870). 



10 Ibid., 1899, ser. 3, t. ii., pp. 297-324 ; t. iii., pp. 125-183 ; t. xi., p. 385. 



" Proo. London Math. Soc, 1906, p. 107; Darboux's Bull., 1906. 



