140 Proceedings of the Royal Irish Academy. 



of zonal, tesseral, or sectorial harmonics multiplied by coefficients which are 

 functions of a, taken with the corresponding constituent of Zn must satisfy 

 the same condition ; so that dividing across by a factor of the form 



(1 - m')' j^s (a*' - ^T (cos 5(^ or sin s0), 



any one h of the 'In + 1 coefficients in Yn taken with the corresponding 

 coefficient h in Zn must satisfy the condition 





,.„»^».-?^%A.J%.A,.?!Lii..o. a) 



Hence h must satisfy the differential equation 



which does not involve a or k. 



It is here that the proper subject-matter of this paper begins. 



Laplace now gives an incorrect method for determining the two arbitrary 

 constants which appear in the solution of this linear differential equation. 

 He writes : — " One of these functions will be determined by means of the 

 function Zn, which has disappeared by the differentiation, and it is clear that 

 it will be a multiple of this function. As to the other function, if we 

 suppose that the fluid covers a solid nucleus, it will be determined by 

 means of the equation of the surface of the nucleus, by observing that 

 ■ the value of Yn relative to the fluid shell contiguous to that surface is the 

 same as its value for that surface." This is not the case : the constants are 

 determined completely by the condition (1) alone, and there is no continuity 

 between the equal density surfaces in the earth and in the liquid. 



In order to prove this, we proceed to find the result of substituting a 

 solution of the differential equation in the condition to be satisfied, and to 

 show that the result is of the form K + K^a^''^^ = 0, where K and K' are 

 independent of a, so that as this condition must be satisfied for all values 

 of a in the liquid, both K and K' must vanish ; and we thus are provided 

 with 4^^ + 2 equation which will determine the 4?i + 2 constants in the 

 final form of Yn. 



In order to obtain the result of substituting a solution of the differential 

 equation in the condition (1), we retrace the steps by which the differential 



equation was obtained from (1). Let b be the known value of a for the outer 



dh 

 surface of the earth, and h^ and h^ the values of h and — at the equal 



pressure surface in the liquid next the earth. 



