350 SCIENCE PROGRESS 



where the coefficients b^, etc., are independent of z and a is any- 

 positive integer or fraction. 



L. J. Mordell {Quart. Journ., 48, 1920) investigates the value 

 of the definite integral 



gat +bt 



1 



ec< -i-d 



dt 



in the general case when a, b, c, d are real or complex, and when 

 path of integration is the real axis or if the integrand has any 

 singularities on that path the real axis suitably indented. 



J. W. Nicholson {Quart. Journ., 48, 1920), in a paper entitled 

 " A Generalisation of a Theorem due to Sonine," proves that 



if m > — - 

 2 



•'0 



where (ai a^ . . . a„) cannot form the sides of a polygon of n sides. 

 The Theorem of Sonine given in 1880 {Ann. Math., 16) was of 



the form ; if m > — - 



2 



f/mM Mbx)J^{cx)x-*^ + ^dx = o 



1 



unless {a, b, c) can form the sides of a triangle, in which case 

 its value is 



[(a + ^ + c){a -\-b- c){c + a- b){b + c - a)Y*-^ 

 [ V 7r23'«-i (w — i^'^^'^c'^J 



The value of the integral with n Bessel functions in the 

 integrand when (ai a-^ . . . a„) can form the sides of a polygon of 

 n sides cannot in general be obtained in a simple manner. The 

 evaluation is performed in this paper for the case when the 

 functions are of zero order, in terms of the complete elliptic 

 integral k. 



This theorem is also the subject of a short communication 

 to the Edin. Math. Soc. {Proc, 37, 191 9) by John Dougall. 

 A proof is given of Sonine 's Theorem based on the Theory of 

 the Potential. 



G. H. Hardy and J. E. Littlewood, in the Quart. Journ., 48, 



