TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 43 



NOTE BY PROFESSOR CAYLEY. 



The object of the paper is the direct verification of the well-known equation 



^n ~ "n¥n 



2 



+ n+l.n SxPn '** Pn '' yy ' C0S ^ 



2 



+ ~ rs n =. <5 2 P .5*.P '• ('V2/') 2 cos 2\lr 



n+2.n + l.n.n-l * n x » v ^ ' r 



where Z„ is the Legendrian function, order n, oixx' + yy' cos t/f, (x 2 + y 2 =l, x' 2 -\-y' 2 =l), 

 and P„, P w ' are the Legendrian functions of the same order of x, x' respectively. By 

 expanding the powers of xx' + yy' cos \p, and in the result the powers of cos xjj in multiple 

 cosines of \fj, the left-hand side is at once thrown into the form 



A + 2A 1 yy' cos \Jr + 2A.^yy'f cos 2i/r + &c. 



where A , A lf A 2 . . . are given rational and integral functions of x, x', y 2 , y' 2 ; and the 

 problem is that of verifying the equations 



A = PJV, A 1 =—± r -8 s P n .S x P n ', ...A,= — ] n /P,(P»' 



n + L.n n + s.n+s—1 . . . n — s + 1 x n * » 



The author starts with the general term A„ writes therein y 2 =l—x 2 , y' 2 =l — x' 2 , and by 

 means of a process (which is of necessity a difficult and complicated one) of the summa- 

 tion of factorial expressions, succeeds in reducing this to the required form, numerical 

 multiple of SX.8^. 



It would probably be somewhat easier (although far from easy) to verify directly the 

 deduced relations 



Ai= — —= — S x &3f Ao, A 2 = — — „ -SxSx'A.i, . . A. s = — —SxSx' A._i , 



n+l.n n+2.n — l n + s.n — s + I 



(in which, of course, y 2 , y' 2 must be regarded as denoting l—x 2 , 1 — x' 2 respectively), thus 

 reducing the problem to the verification of the first equation 



Ao ^r n r n ; 



and as regards this equation it would be perhaps easier (instead of writing 

 y 2 =l — x 2 , y f2 =l — x' 2 ) to homogenise the equation by introducing in the several terms 

 thereof the proper powers of x 2 + y 2 , x' 2 + y' 2 . The left-hand side would thus be a given 

 function homogeneous of the order n in x, y, and homogeneous of the same order in 

 x', y' (y, y' entering in the squares y 2 , y' 2 respectively), and this should be identically 

 equal to the right-hand side PJP/ expressed in the known form 



/ _ n.n—1 „ „ „ , n.n — I. n— 2.n— 3 n , , „ V .,„ n.n — 1 ,, 



■ x" 



„ „ n.n— l.n — 2.n— 3 „ . d D V »« n.n — 1 , „ „ , . \ 

 ■¥+ 214^ »"-V- &c.j(aj'» ¥ -x' n - 2 y 2 + &c.j 



V 2 2 



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



