SECTIONAL TRANSACTIONS.— A. 299 



Prof. E. C. TiTCHMARSH. — Functiotis which are their own Fourier Trans- 

 forms. 



The Fourier reciprocity between two even functions f(x), g(x), is expressed by 

 the formulae 



CO CO 



g{x)== '\/{-Mcosxyf(ij)dy , f(x)= ^(^\\ cos xxjg{y)dy. 

 In some cases g(x) = f{x) ; this is true for example for the functions 



JL , e-*^' sechx\/M- 



\'x ^ ' 



We find that there are formulae which give general expressions for all functions which 

 have this property. The simplest such formula is 



C-\-ico 

 C — iao 



where ^{s) =\\i{l — s). 



Mr. T. W. Chaundy. — A Note on the Hypergeometric Equation. 

 Equations of the generalised hypergeometric type : — 



f(xl>) y=x'"g{xl>) y 



are classified by the residues (mod m) of the zeros of /, g. Conditions are obtained 

 that this equation be soluble by algebraic and logarithmic functions, and that the 

 equation 



f(xD) y=xV'.y 



be soluble (1) by exponential functions and polynomials, 

 (2) by Bessel functions. 



The theory, due to Mr. J. L. Burchnall and the author jointly, yields various well- 

 known results and also certain formulae (possibly new) for the solutions P„, Q„, of 

 Legendre's equation. 



Dr. D. M. Wrinch. — Recurrence Relations and some Definite Integrals 

 involving Legendre Polynomials. 



It is shown that the integrals, with regard to \i of the functions 



P„(t^) ^ P,. (!i.)Pm([i .) ^ lJlP„((J.)P„,(i^) 

 V — [x' V — fi. ' V — (i 



between the limits — 1 and + 1 follow readily from the recognition that these 

 definite integrals, as functions of ?i, satisfy the familiar recurrence relation of P„. 

 The values of the integrals of the functions 



V„-{yi.)P.,{[l) ^ txP,/(iJl)P,„(il) ^ (^^ - l)P,/(tJL)P„/(tJL ) 

 V— (i. ' V— [X ' V — (X 



between the same limits, and of some allied integrals, are then inferred almost 

 without calculation. 



Miss R. C. Young. — The Algebra of Infinities. 



This paper is concerned with the manipulation of ( positively and negatively 

 ' infinite values ' as thej' occur in analysis (i.e. by passage to a limit). 



