346 REPORTS ON THE STATE OF SCIENCE. 



equation which has been called by Poincare the Laplace transformed 

 equation. 



Laplace extended his method to definite integrals of the type 



[t>-'<t>{t)dt = g(z) .... (2) 



and made considerable use of the method in his researches on probability. 

 It will be convenient to call integral equations of the type (1) and 

 (2) integral equations of Laplace's type. The term integrals of Laplace's 

 type is sometimes given to integrals of the form 



\cos(xt)<f>(t)dt, 



but it is more convenient to name these after Fourier. 



The integral equation (1) was studied by Abel, 1 who obtained a 

 number of properties of Laplace's transformation. Abel also proposed 

 the problem of solving an integral equation of the first kind, and stated 

 Shat he had obtained a solution for a general type of equation. 2 The 

 equation (2) was studied as an integral equation for determining <j>(t) 

 when g(t) is known explicitly by Eobert Murphy 3 in 1832-1834. He 

 gave the formula 



<£(0 = coeff. of - in g{z)t~ x 



6 



8 



and considered the general problem of determining a function x(<) for 

 which 



o = f|»- x (t)dt . . . . (3) 



when z has the values 1, 2, . . . n. The last problem had been treated 

 previously by Jacobi, 4 but not solved completely. It was shown by 

 Liouville 5 that the above equation cannot be satisfied for all positive 

 integral values of z when \{t) is a function which only changes sign a 

 finite number of times. This result is of importance in settling the 

 question of the uniqueness of the solution of an equation of Laplace's 

 type. The result was extended by Lerch, 6 in 1892, to the case in which 

 x(t) is continuous, and has been extended to other sequences of values of 

 z and other types of function x(t) by a number of subsequent writers. 7 

 It has been shown by Kluyver and N ielsen that equations of type (2) 

 are of great importance in expansions in series of inverse factorials. 

 The integral equation 



f(x) = U*<f>(t)dt .... (4) 



1 (Euvret, t. 2, p. 67. 2 Ibid., t. 1, p. 11. 



s C'amb. Phil. Trans., vol. iv., 1833, pp. 353-408. 



4 CreWs Journal, vol. i., p. 301 (1826). 5 Uouville't Journal, t. 2, p. 1, 1837. 



8 Acta Math., 1903, p. 339. 



' Stieltjes (1893), Correspondence de Hermit e et Stieltjes ; Landau, Fend. Palermo, 

 1908 ; C. N. Moore, Bull. Amer. Math. Soc, 1908 ; W. H. Young, 1910, Mett. Math., 

 vol. lx. 



