514 Prof. Thomson on the Uniform Motion of Heat. 



The integrals of these expressions, between the hmits «, = 

 and <7j = rt'j, are the components of the attraction of an elhpsoid 

 whose semiaxes are a\, b\, c'„ or a\, «', \/(l — e^), a.\ \/ (\ — e'"^) 

 on the point {x, y, z). Now by (1) we may express each of the 

 quantities h, c, b^, Cj in terms of a and o,, and the equation 



^2 y2 „2 ,j,2 y% „2 



^ + i^ + ?=^' °''^+a2._,2«^2 +«2_,/2«^2=l • (3) 



enables us to express either of the quantities a, ff, in terms of 

 the other. The simplest way, however, to integrate equations 

 (2) will be to express each in terms of a third quantity, 



M=^ (4) 



a 



Eliminating a from (3) by means of this quantity, we have 



-2 



<=:mV+-J1— 2 + 



-2_e/2• 



Heuce 



a,da, = ^ux^+ !, j^ + ,V_:^2.. }du 



[u-^-e^f (M-2-e'2)2 



(^2 y2 „2\ 

 "4 + 74 + '^)a^^u-'^du = a^'^p--u-^du. 



Also from (4) we have a = —; from which we find, by (1), 



6 =s ^ / (1 -- e'^u'), c = ^ i/ (1 - e'^-u") . 

 u u 



By (1) also, b^ = ay\^{\-e'^), c^=a^\/{\-e'^). Making these 

 substitutions in (2) and integrating, we have, calling a' the value 

 of a when «j = tt'„ 



A=47ra;i/(l-e2)-v/(l-e'2)(«'- " "'^ 



Jo 



B=47ryi/(l-e2)^(l-e'2) p^ 

 Jo (1 — 



C = 47rzi/(l-e2)\/(l-e'2) ^ 

 Jo 



V'(l-eV)'v/(l-e'V) 



(l-eV)*(l-e'V)5 

 u^du 



L(5) 



(l-eV)*(l-e'V)2 



If the point atti-acted be within the ellipsoid, the attraction of 

 all the similar concentric shells without the point will be nothing, 



and hence the superior limit of u will be the value of — at the 



a 



surface of an ellipsoid similar to the given one, and passing 



through the point attracted. 



Now in this case Oj = a, since a is one of the semiaxes of an 



