in Homogeneous Solid Bodies. 511 



Now by {b) dn^ — , and hence 



Integrating this^ we have 



The two constants, k^ and C, must be determined by the con- 

 dition i) = Vj when a^a^, and r; = when a = cc ; the latter of 

 which must be fulfilled, in order that the expression found for v 



may be equal to / / -^ — - . 



To reduce the expression for v to an elliptic function, let us 

 assume 



a =/cosec 4^ \ / />\ 



flj=jcosec 9iJ 



which we may do with propriety if /"be the greater of the two 

 quantities / and ff, since a is always greater than either of them, 

 as we see from {d). On this assumption, equation (e) becomes 



'—f J„./(l-c'^sin^</>)+^- —f ^^"^ + C, 



where , g 



^=1 ^^ 



Determining from this the values of C and ^j by the con- 

 ditions mentioned above, we find C=0, and 



k - ^""^ ■ (h\ 



hence the expression for v becomes 



^=^^fe^, W 



The results which have been obtained may be stated as fol- 

 lows : — 



If, in an infinite solid, the surface of an ellipsoid be retained 

 at a constant temperature, the temperature of any point in the 

 solid will be the same as that of any other point in the surface 

 of an ellipsoid described from the same foci, and passing through 

 that point ; and the flux of heat at any point in the surface of 

 this ellipsoid will be proportional to the perpendicular fi-om the 

 centre to a plane touching it at the point, and inversely propor- 

 tional to the volume of the ellipsoid. 



This case of the uniform motion of heat was first solved by 



