commonly called the equation of Laplace's functions. If we put 

 sin (- n cos &=&, then the equation (2) may be written 



and the operation ta n possesses the following property, namely 



w_ n or rt + w 2 =w ra _i *?_(_,) + (n I) 2 ; 

 hence it is easily shown, that in general the complete solution of (2) is 



where w is the solution of 

 namely 



- ^T tan -- 



and the operation rov^re-i ... ra^Wi is easily seen to be equivalent to 

 (sin 6) -" (sin -^ sin J ". 



(This result is compared with that obtained in a different way by Pro- 

 fessor Boole (Cambridge and Dublin Journal, vol. i. p. 18), to which 

 it bears a general resemblance, but the author has not succeeded at 

 present in reducing the one form to the other.) 



In the case in which u n does not contain $, we have 



The general expression for a " Laplace's coefficient " of the wth order, 

 not containing ^>, is therefore (sin 0)- ra rsin sin J . C ; and if 

 this be called v n when C= 1, the development of (1 2rcos 0+ r*)~* is 



and it is shown that the coefficient of ^ - in the development 



of (l- 



(sin 0) ~* ~ ' ( sin ^ sin J (sin 0) 4 . 



