392 REPORTS ON THE STATE OP SCIENCE. 



The case in which fi(x) and y(x) are solutions of a differential 

 equation is of special interest, for then h(x,t) is generally a Green's 

 function (section 8). The conditions in which two solutions of the 

 differential equation exhibit the required properties are indicated by 

 the following theorem due to Dixon, Weyl, and Kneser. 



Let 



L( " )S l[^§]-*»"=° 



bs a lin3ar differential equation in which p(s) is a positive function 

 having a continuous first derivative for s>0, while q(s) is continuous 

 for all positive values of s and such that 



q(s) > 0. 



Loi y(s) bo a solution of L(?<) = defined by the initial conditions 



y(°)> iA°)y'{°) = ! ; 



then y > 0, y(s) > 0, for s > 0. 



tt>S 



Similarly, if j3(s) ig a solution defined by the initial conditions 



u{o) = 1, u{l) = 



wo havo 



W < 0, ft> 0. 

 as 



Hence the solutions ft(s), y(s) satisfy the above Conditions. 



It has been proved by Mercer (I.e.) that the necessary and sufficient 

 conditions that a function g(s,t) may b3 of positive type are that the 

 quadratic form 



2 2 gtp,q)x P X q 



V 1 



should be definite l for any choice of a set of points p,q ... in the 

 given interval. This theorem has been extended by W. H. Young. 



It is clear that the properties of definite functions may be extended 

 to functions of several variables. It is important to notice that the 

 functions 



1 1 



[(x -xy + (2/ - y l f + (2- *')»]» [{x + V) 2 + (y - yf + (* - *>)*]» 



in which the variables (x,y,z) (x x ,y^,z) are the coordinates of points on 

 a closed surface are definite functions. 



The first function occurs in potential problems connected with a 

 single closed surface, the second function occurs in potential problems 

 connected with a closed surface in front of an infinite plane. This 

 property of the functions depends on the identities 



1 Tho conditions are that the quantities ff(tt t s) <j (****}> g( S,S ' ll>3 \ ■ ■ ■ • should 



be positive ior every set of values of the variables s^s^u .... contained in the 

 interval (a,b). 



