338 Expansion ofF (fi, w) in a series o/'Laplace's Coefficients. 



efficients every term but tlie first vanishes; and, since Po=l» 

 we have 



Multiply both sides by F (/x, w) and divide by 4 tt, and ob- 

 serving that the function may be written inside the signs of in- 

 tegration, because ^ and w are variables independent of /t' and 

 w', we have 



But, by what has been shown above, the whole quantity under 

 the signs of integration vanishes, except when //-=/"'' and a; = a)', 

 and it vanishes even for values of //,' and co', differing by the 

 smallest infinitesimal quantities from these. Hence the equa- 

 tion may be written 



But Pj-, and therefore the general term of this series, since 

 the operations performed on the terms are linear, is a function 

 of yu-and w, which satisfies Laplace's equation. Hence F(/^,c«) 

 can be expanded in a series of Laplace's coefficients. — Q.E.D. 



N.B. It cannot be objected to this theorem, that because a 

 series is made use of the value of which becomes infinite under 

 certain circumstances, therefore the results are not to be de- 

 pended upon. For the integration modifies each term of the 

 series, and its sum is not then infinite, as we have seen indeed 

 in the particular case of the first integration in the demon- 

 stration. 



Suppose Fo+ Fj + ... + Fj-f is the series into which F(//, w) 

 can be expanded. Then* 



•^0^"')=iX\X^'{Pon+ SPiF, + ... + (2/+ l)PiF',.+ ... \diJd^\ 



If the number of terms in F (ytt, w) is finite, and Fj is the last, 

 then it is evident that the terms involving Pj_li ... are all de- 

 stroyed. But suppose the number of terms in F (/it, w) is in- 

 finite, then, since by hypothesis F (/a w) does not become infi- 

 nite, the series must converge, and the term F^- must become 

 infinitely small as i becomes infinitely great ; and the integrated 

 series on the right-hand will converge and not become infi- 

 nite. 



* Pratt's Mechanical Philosophy, Second Edition, p. 165. 



