Expansion qf'F (n*, w) in a series o/" Laplace's Coefficients. 337 



Petit, with regard to simple atoms, is capable of a greater 

 degree of generalization than we have hitherto been inclined 

 to admit. 



IjIV. Demonstration of the proposition that every function 

 F (jM,, cu) nxihich does not become infinite betiveeji the limiti?ig 

 values — 1 a7id 1 q/'ix. atid and It: of w may he expanded 

 in a series o/" Laplace's Coefficients. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 

 'VT'OU will not perhaps think the following concise demon- 

 -*• stration of this most important proposition unworthy of 

 your consideration, based though it be on demonstrations al- 

 ready before the public. I am, your obedient Servant, 



Procul. 



Let [LffJ + V\ —jj!'^ Vl— /jl^ cos {oo' — co) = p; 

 and suppose that 

 {l+c^-2cp)-^=.Vo + PiC + PgC^ + + P/ + 



then it is known* that Pq, P^ are some of Laplace's co- 

 efficients. Differentiate with respect to c, multiply by 2 c, and 

 add the result to the above, and we have 



^~''' =Po + 3PiC+ + (2f+l)P,.c^l+ 



{l+c^-2cp)i 



When c = 1 this series equals s:ero, except when j9 = 1, in 

 which case it appears to be indeterminate ; but it is easy to 

 show that in that case it equals infinity^ for each of the coeffi- 

 cients Pq, Pj becomes unity when p= If. It is not dif- 

 ficult J, moreover, to show that when j3=l, a;=a)' and /a=//-'. 

 Hence we arrive at this remarkable result, that the series 



Po + 3P, + 5P2+ + (2/+l)P^+ 



equals zero for all values of the variables, except when /* = /x' 

 and «) = »', in which particular case it equals infinity. The 

 series is therefore discontinuous^ and will destroy any finite 

 function by which it is multiplied, except for those particular 

 values of the variables ; it will destroy the multiplying func- 

 tion even for values of the variables differing by the smallest 

 infinitesimal quantities from those which satisfy the conditions 

 yu,=ft' and w=«)'; this is because the value of the series is dis- 

 continuous, and per solium passes from zero to infinity. 

 Having premised thus far, if we integrate the series as it stands, 

 term after term, then by the known § property of Laplace's co- 



* Pratt's Mechanical Philosophy, Second Edition, p. 161. 

 t Ibid. p. 160. X Ibid. p. 166. ^ Ibid. p. 165. 



FhiU Mag. S. 3. Vol. 25. No. 167. Nov. IS*^. Z 



