400 Prof. A. Gray on the Attraction of 



which, if we write u' for v/fi and substitute, becomes (accents 

 omitted) 



rx IpS g— )du 



Y^Trpabci / \ (r ' { -" j . . (24) 



Again, as may easily be verified, 



\ 2 — 7rpa 



Jm Jo \/ {a 2 ^-.uj(b 2 + u)(c 2 -{-u j 



= iT P ahc(l- h L) P— .---■ — ! d - U • . (25) 



Jo v / (« 2 + ^)(^ + ^)(c 2 + ^) 



Hence 



r . (l-2-f— V w 



Jo v/(a s -t-tt)(6 s +.i0(c s + u)' ( " 



6) 



The value of V given by (20) and (26) was verified by 

 Dirichlet by showing that it satisfies Poisson's differential 

 equation 



d 2 V d 2 V d 2 V 



<--., +^ +°-., = -4t>7> . . . . (27) 

 c^ 2 d/r <3" r v y 



within the attracting matter, and Laplace's equation elsewhere, 

 gives continuous values of the force-compomnts —dV/da, 

 —dYjdy, —dV/dz at the surface, and vanishes for w=oo . 

 Thus (20) and (26) give the solution of the differential 

 equation of the potential for the given distribution of matter, 

 and the known family of equipotential surfaces possessed by 

 each homoeoidal part. As has already been remarked, it was 

 shown by Lame that the differential equation could be inte- 

 grated in these circumstances. ■ It would, however, be outside 

 the scope of the present paper to enter into a discussion of 

 the process. Suffice it to say that any solution which fulfils 

 the conditions indicated above can be proved to be the only 

 one. 



21. The lemma stated in § 14 above enables the whole 

 problem of the ellipsoid to be disposed of very simply ; but, 

 so far as merely proving Poisson's theorem of the direction 

 of the attraction of a thin homoeoid is concerned, nothing more 

 elegant has ever been invented than the demonstration 

 published in Crelle's Journal (Bd. 12, 1834) by Steiner, 

 immediately after the theorem was announced by Poisson in 

 the memoir of 1833, to which reference has already been made. 

 Steiner's construction is shown in the adjoining diagram. 



