10 PROCEEDINGS OF THE AMERICAN ACADEMY, 



where $ represeuts the quantity 

 U V. - V) sinh {^/FT^') - ( n - V) sinh ( - <' - '>^^ ] 



sioh ( 2^V/>2 + qA 



and where /? and q are integers. 



F is evidently the temperature on Fourier's hypothesis within the 

 parallelepiped, if its bases and sides are kept at the temperatures F^, F„ 

 and V respectively, when the How is steady. In this case the specific 

 conductivity of the material of which the liomogeneous parallelepiped is 

 made does not aftect the temperatures within the solid, and the lelative, 

 not the absolute, dimensions of the parallelopiped are of importance. Tiie 

 interpretalion of the equation (21) when/(^) and 6 are assumed to be 

 different is obvious. 



(3) A function V, which involves the time and the distance from the 



9V 9V 9'V . . 

 co-ordmate plane z ■=. 0^ is contniuous, as are -7^ , -,^ , -?r-\ , in tne 



at d z d z^ 



region It, bounded by the planes 2 = 0, z = I. Within R, V satisfies 



9 V ,9- V 

 the equation -^^-~ = <r ^ ., • V vanishes when z ^ I, and has tlie 



d t d z- 



constant value J'^ when c = 0, whatever t is. If, when < = 0, F= Fo^(c) 

 for all points within R, 



X[Vw + '-l]»i"^rfA]. (22) 



If <^ (z) has the constant value c, 

 V= Fori-~^ + ^|(2c- l)[r'-sin'^+ vr'^'sin^ + 



ir>sin^ + ...]-[U"-sin^+le--sinip + ...]M, (23) 



where T=P /a'^TrK 



Equation (23) would give, on Fourier's assumptions, the temperatures 

 at any time within a homogeneous infinite plane lamina of thickness I 

 initially at the uniform temperature c V^, if, from the time ^ = 0, one face 

 were kept at the constant temperature Fq and the other at the constant 

 temperature zero. 



