ON THE THEORY OE INTEGRAL EQUATIONS. 355 



earthquake wave at different points of the earth on the supposition 

 that the earth is symmetrical about its centre. This problem has been 

 solved by G. Herglotz 1 and by the author. 2 The method has been 

 applied to some recent observations by Wiechert and Geiger. 3 



Abel's equation has also been used by Boggio 4 to solve a differential- 

 integral equation which occurs in the problem of the motion of a sphere 

 in a viscous fluid. This equation has been obtained by Basset 3 and 

 Picciati, 6 and was solved by the latter by a method of expansion. The 

 results are of some physical interest, as it is proved that the motion of 

 the sphere tends to become uniform, and then its limiting velocity is 

 given by Stokes's law. A good account of the theory is given in a 

 recent paper by Basset, ' Quarterly Journal ' (1910). 



5. Integral Equations with Variable Limits. 



The idea of an integral equation in which a variable quantity occurs 

 in one of the limits of the integral appears to have been first considered 

 by Babbage, 7 although no method of solution was suggested. 



Cauchy 8 in (1815) solved a special integral equation of this type by 

 differentiation, but the first general method of solution appears to have 

 been used by Poisson 9 in 182G. He obtained an equation of the form 



g(t) = 1it)-% k \f'(t-0)m^ ... (i) 



o 



in which the unknown function </< appears both inside and outside the 

 sign of integration. An equation of the form (1), in which the kernel 

 is of the iovmf'(t — B), will be called an integral equation of Poisson 's 

 type. Poisson solved the equation by expanding </.(/) in powers of k, 

 thus obtaining an infinite series of multiple integrals, but he did not 

 establish the convergence of the series. 



This step was supplied by Liouville 10 in (1837). He arrived at the 

 integral equation 



g(t) = +(t)-\L(t,<))<l>(t>)M .... (2) 



in his classical researches on linear differential equations and the expan- 

 sion in the so-called oscillating functions. These functions satisfy a 

 differential equation of the Sturm-Liouville type, viz., 



d f dV\ 



which occurs in the theory of the conduction of heat in a heterogeneous 

 bar and in many other problems of mathematical physics. The set 

 of functions is obtained by taking the different possible values of the 



1 Phys. Zeitschr., 1007. - Phil Mag., 1910. 3 Phyt. Zeitsehr., 1910. 



4 lic'/id. IAneei, 1907. s Hydrodynamics, vol. ii. G fiend, Lined, 1907. 



* Memoirs of the Analytical Society, Preface, 1813. 



8 Mt'-mti'ire twr hi thSorie den (Judex. 



9 Memoirs sur hi thSorie da maqvMitvie en movrement. (Kuvres t. :!, No. 5, 

 pp. 41-72. '" Liouville's Journal, vol. ii. 



