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XC On Products of Legendre Functions. By J. W. 

 Nicholson, F.R.S., Fellow of Balliol College, Oxford*. 



TIIHE fact that it is possible to express the product of the 

 JL two functions P n (ji), Pm(/*) as a series of functions of 

 the type P r (/<0 was discovered independently by Couch 

 Adams and Ferrers many years ago. Apart from a paper 

 by Sir William Niven, the subject of these products, which 

 can be developed in a manner of some significance for 

 applied mathematics, does not appear to have attracted the 

 attention which it deserves. Niven's paper made a notable 

 advance, being essentially more general in its scope than the 

 intuitive method of the earlier investigators — afterwards 

 arranged more concisely by Todhunter. But no author has 

 considered the equally interesting and important products 

 in which functions of the type Qn(/^) appear, though these 

 can be included in an investigation by a method which is, 

 after Poincare, now familiar in the theory of differential 

 equations. An outline of this investigation, with some of 

 the more important properties of the functions, is our 

 immediate object. 



Let an accent denote a differentiation with respect to /x, 

 and let P stand for P ?l (/u,) or Q„(/x), and Q for ~P m (ji), Qm(ft). 

 For convenience, we write also M = m(mf 1), N = ?2(n + 1). 

 Then 



(l-/. 2 )P /, = 2 / .P'-NP, 



(1_^)Q'' = 2^Q'-MQ. 



We first form the linear differential equation of the fourth 

 order satisfied by PQ. Its general solution must be 



A P n P m + B r n Q m + G P w Q n + D QmQn; 



where A, B, 0, D are constant , and from its solution in the 

 form of series we can derive the four typical products. 

 If 



y=PQ, 



we have 



y = 2P'Q' + PQ" + QP" 



=2P'Q' +f |^(2 /i Q'-MQ) + r ^ ? (2 /i P'--NP), 



whence 



(l- M %"-2 w ' + (M + N)y=2P'Q'(l-^). 



* Communicated by the Author. 



