PEIRCE AND VVILLSON. — THERMAL CONDUCTIVITIES. 9 



— 9 V 



where Fis a constant, and ^ — represents the derivative of V taken in 



d n 



the direction of the exterior normal. If the origin of rectangular co- 

 ordinates be taken at the centre of the lower base while the axes of 

 X and y are parallel to the sides of this base, V is given by the equation 



P ^ CO k =L 1X1 



v_ ' 



p ^ CO a; =: CO 



V +^(^j, • COS (n^ y) 2j'k • cos («^. x) O , (20) 





where O represents the quantity 



Here ni, ??2, ns, etc. are the successive roots of the equation 



K n . tan {n d) ^ A, 



and Ay,, h stands for the radical ^,j 2 _|_ ^j^/i, while Cj, c^, C3, etc. are the 

 coefficients of the successive terms in the development, 



1 = d cos (?^l d) + Co, cos (wo 0) + Cg cos (??3 ^) + 



so that t•^• = 4 sin (?^^^•a) -^ (2 nua + sin (2 rika)). 



It is to be noticed that equation (20) would give, on Fourier's assump- 

 tions, the final temperatures within a homogeneous parallelopiped of spe- 

 cific internal conductivity k, and of external conductivity h, if the lower 

 base were kept at the constant temperature Vq and the upper base at the 

 constant temperature F„ while the sides were exposed to the atmosphere 

 at the temperature V. In this result the absolute dimensions of the 

 paralleIopi[)ed are inextricably involved with the value ot h / k. 



(2) The square bases of a rectangular parallelopiped of height / are 

 2 a long and 2 a broad. A function, F, harmonic within this parallelo- 

 piped, has the constant value Vq at the lower square base, the constant 

 value Vi at the upper base, and the constant value F on the other faces 

 of the parallelopiped. If, then, the centre of the lower base be used as 

 origin of co-ordinates, with axes of x and y parallel to sides of the base, 

 Fis given by the equation 



<'^^)-<^>*-(-) 



