1879.] Conduction of Heat in Ellipsoids of Revolution. 101 



(zero) temperature, X is given either by 0=Q^(Xfe) or 0=Q z m (\a), 2a 

 and 2b being the major and minor axes of the ellipsoid. The latter is the 



• • G 



more convenient for expressing X in powers of the eccentricity e= ~- 



It is shown how the values of X may be found, and in particular for the 

 first root of m=Q, the equation becomes 0.^=0, which gives for X, 



3 135V 7 J 

 When the solid cools by radiation the boundary condition is 



^ + ^v/^ 2 V=0(f=X&); 

 X 



and, to deal with this condition, it is necessary to show that any func- 

 tion of v, at least every function capable of expansion in the form 



■SAjjCOO, may be expanded in the form 2B„0 m . The possibility of 



m 



this depends on the theorem j ' O^0^dv=O, z not ■=*', which is proved 

 true. 



In particular the coefficients of the expansion of v~9 m in this form are 

 fully worked out. 



To satisfy all the conditions of the problem we must now assume for 



V an expression of the form (cos m<fi or sin m0) e 2C r #^Q^ r ; and 



general equations are given whereby both the values of X and the 

 coeflicients (C) are determined. If we require the expansion of X only 

 as far as e 4 , the equation which finds it can be readily given. If we 

 wish only X up to e 2 , this equation breaks up into a system of the form 



in which cf r are constant coeflicients whose values are found. The 

 course of the subsequent approximation is free from difficulty. 



To calculate the coefficients which depend on the initial state of the 

 ellipsoid, we have to evaluate the integral 



J -* Jo 



in which v may be either 0«G* or SCrfl^Q^", as the case may be ; and 

 to do so we must determine the folio wins' single integrals : — 



J-l J-l 



" l) * 



