404 Dr. J. H. Jellett on Laplace's Equation. 



and when p = 2q + 1, the order is 



4^ + 4 + 2(m—0 — 1) + 2 + l = 2q + 2m + 5 = 2q + n + 6. 



The term retained in the value of v, (6), corresponds tojo = 0. 

 The order of this term is therefore = order of a + n + 4, when 

 is small. Hence \in +4 be not < 0, the order of this 

 term must be = or > order of a. The order of the next term 

 will = order of a 2 + n + 6, and the orders of all succeeding 

 terms will be higher when 6 is small. The degree of the 

 approximation is therefore as has been stated. 



The same reasoning which has been applied to the potential 

 will hold also for the resolved attraction. In fact, as we have 

 seen, when 6 is small the orders of the corresponding parts of 

 the potential and of the resolved attraction are the same. 

 Hence : — 



If quantities proportional to the square and higher powers of 

 the thickness of the shell be neglected, Laplace's equation is true 

 for all positive values of n, and for all negative values which do 

 not numerically exceed 4. 



It will be observed that in the foregoing investigation I have 

 taken Laplace's equation to refer, not to the potentials and 

 attractions of the solids, but to the potential and attraction of 

 the shell included between them. It is upon this consideration 

 that the validity of the proof of Laplace's equation depends 

 for negative values of n numerically greater than 2 and not 

 numerically greater than 4. For such values of n the differ- 

 ential expressions for the attraction and the potential admit of 

 a true integration for the shell, or its equivalent the material 

 distribution. But these expressions do not necessarily in such 

 cases admit of a true integration for either of the solids. Thus, 

 if n be numerically greater than 2 and not greater than 3, the 

 differential expression for the attraction does not admit of a 

 true integration ; and if n be greater than 3, neither expression 

 admits of a true integration. In the former case the attrac- 

 tion is infinite, and in the latter both attraction and potential 

 are infinite. 



Hence it is evident that the true cause of the failure of 

 Laplace's equation (when it does fail) is that the quantities 

 with which it is concerned cease to be finite. It does not fail 

 for such a shell as Laplace describes, because the thickness of 

 this shell is zero at the attracted point and exceedingly small 

 in the immediate neighbourhood. It results from this con- 

 struction of the attracting shell, that the potential and resolved 

 attraction remain finite for higher inverse laws of force than 

 they would for an ordinary solid. 



