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II. — On the Transformation oj Laplace's Coefficients. By Dr Gustave Plarr. 



(Read 2nd December 1889.) 



INDEX TO CONTENTS. 



PAGE 



Introduction, 19 



Section I., 20 



,. n., 24 



PAGE 



Section III., 29 



„ IV-, 34 



V., 38 



The development of the inverse square root 



Z = (l-2«s + a 2 )-* 

 into a series 



gives rise to the coefficients Z,„ which have been called "Laplace's Coefficients." 

 If in Z n we substitute for z the expression 



z = xx' + yy' cosxjs , 

 where 



x' 2 + y' 2 = l, 



then the function of i/>, which Z„ will represent, can most appropriately be expressed by 



Z„ = 2 cos (st/t) . N, . 



The object of this paper is to show, by actual calculation, that the coefficients N, 

 (functions of x, x', y, y') can be worked out by elementary algebraical processes, the only 

 auxiliary taken from higher analysis being the expression of the powers of cos \ff in 

 function of the cosines of the multiples of i//. 



As to our notations, we shall observe throughout the following restrictions : — 



1. The function U(x) shall not be employed otherwise than for integer positive 

 values of x, so that 



n(cc) = l .2.3 x=(x!) 



2. The factorial 



'/'■ 



a . =a(a + r) .... (a + t — lr) 



will be made use of for integer values only of r (as, for example, r= + 1, or r= — 1. or 

 r= +2, &c), the exponent t being integer positive. 



VOL. XXXVI. PART I. (NO. 2). E 



