408 REPORTS ON THE STATE OF SCIENCE, 



is satisfied by a definite integral of the form 



CO 



V = (" e- u J m (f,t)j>(t)tdt. 



We may determine </>(£) so that V =f{i>) when z = by solving Hankel's 

 integral equation ; we may (when m = 0) determine <p(t) so that 

 V = F(,?) when p = by solving an integral equation of Laplace's type. 

 The solution of one equation may be made to depend on that of the 

 other when the solution of the partial differential equation for the given 

 conditions is known. The differential equation 



9 2 V 1 f)V n 2 V 

 dft l p dp vz 2 



is satisfied by 



v _ 1 f s i n ksjr 2 - 2aru + a 2 , . „ 



_ f sin k-Jr 2 — 2arfj + a 2 



J Vr 1 — 2aru + a 2 AV ' 



(1) 



(2) 



and 



'2tt 



V = ~ f *"»*-/(* + ip coaa)da . . , . (3) 



Air J 



w 



CO 



V= fr' : J„( pv / ?TF)^)(l« . . . (4) 



o 



where z — r^, p 2 + z 2 = r 2 . 



If these expressions all represent the same solution we have when 

 P = 



CO 



1 f sin k{z - o) 



— CO 



J 3 — a 



V 



CO 



/(*)= f «— ,/,(«,)<?« 



These integral equations are of the type considered by Hardy. 1 He 

 has shown that \if{z) is defined by an equation of the second type with 

 m written in place of k, then the first equation is satisfied by q>(a) =/(a) 

 if k >m. This result indicates that a solution of the partial differential 

 equation that can be expressed in the form (2) can also be expressed in 

 the form (1). By putting z = in equations (1), (2), (3), (4), the 



1 Proa Land. Math. Soc. ser. 2, vol. vii. (1909), p. 445. 



