SECTION IV. 



ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS OF THE 



Jm p tJP P 



FORM ^4" x^. 



d/j, m dp,P 



1. WE have already shewn (see Vol. I. p. 487) how to exhibit the 

 product of two Legendre's coefficients, P n P q , by means of a series of 

 Legendre's coefficients. In order to complete the theory, we must shew 

 how to multiply together any two Laplace's coefficients, so as to exhibit 

 the product as a sum of Laplace's coefficients. There can be little doubt 

 that a method similar to that which has been already employed will be 

 equally successful in the more general case. 



The general form of two Laplace's coefficients, whose product we wish 

 to express, may be denoted by 



1m p m 



and 



where X is the longitude. The product is of the form 



d m P dP 



