410 REPORTS ON THE STATE OF SCIENCE 



an equation of this type can be satisfied may be obtained by considering 

 the homogeneous integral equation of the second kind. 1 



•In 



and giving k- the different possible values for which the functions /*(*.) 

 are zero. These functions may be determined by a method of successive 

 approximations by expanding both sides in powers of k. 



The applications of Predholm's equation to partial differential 

 equations of elliptic type are very numerous. We must refer the reader 

 to Andrae's dissertation * and papers by Picard 3 and HUbert. 4 Equations 

 of hyperbolic and parabolic type have also been discussed by Le Eoux, 

 Andrae, Holmgren, Goursat, Picard, Hadamard, Lauricella, Levi, 

 Mason, Myller, and many other writers. A short accouut of the appli- 

 cations is given in Horn's ' Partial Differential Equations.' 



23. Applications to Problems in the Theory of Elasticity. 



The problems connected with the equilibrium of an elastic body 

 when either the displacements or tractions are -given over the surface 

 has been reduced in various ways to the solution of a system of equations 

 of Predholm's type. 



E. and F. Cosserat 5 effected this reduction in 1901, and solved the 

 equations by a method of successive approximations. 



Predholm 6 applied his method of solving the integral equation of 

 the second kind to show that the integrals of the differential equations 

 which determine the equilibrium when the surface tractions are known, 

 are meromorphic functions of the elastic constant. The theory has 

 been discussed very thoroughly by Lauricella, 7 Marcolongo, 8 and Boggio ; 9 

 the last author makes use of Green's functions. 



The subject of the Prix Vaillant for the year 1907-08, proposed by 

 the Paris Academy was that of the equilibrium of an elastic plate with 

 fixed edges. The analytical problem is that of determining a solution 

 of the equation 4 



?•* = **+**> +*%=f(x,y) 

 dx i ox-vy 1 dy* 



under the conditions that u and its normal derivative along the boundary 

 are zero. The prize was divided between Hadamard, 10 Lauricella, 11 

 Korn, 12 and Boggio. The same problem has also been discussed by 

 Marcolongo, 8 Zaremba, 13 and A. Haar. 14 The mathematical investiga- 



1 Meu. Math., April 1910. 2 Gottingen, 1903. s Rend. Palermo, 1906. 

 4 Gottingen Nachriektm, 1906. 5 Comptts rendus, t. 133 (1901). 



6 Arkiv for mathematik ochfysik, Bel. 2, n. 28. 

 ' II A'uovo Cimento, 1907 (5), vol. xiii. 



8 Rend. Lincei, vol. xvi., ser. 5 ; Toulouse Ann., 190S. 



9 Bend. Lincei, 1907, pp. 248, 441. 10 Mhnoires de VInttitut, 1SC8. 

 " Bend. Lincei, 1907; Acta Mathematics, 1908-09. 



" Annates de VEcole Normale Supericurc, ser. 3, vol. xxv. (1908). 



13 Ibid., vol. xxv. (1909), p. 337. 



14 Gottingen NachricMen, 1907. 



