PROFESSOR TAIT ON THERMAL AND ELECTRIC CONDUCTIVITY. (2y) 
second gives an infinite number of real positive values of m, say m,, m,, &c., 
in ascending order of magnitude, in terms of 7, (the radius of the cylinder), 
k,and h. Now his easily found (approximately) from the rate of cooling, and 
kis known. Hence we determine the values of m, and have 
0 = Aye, + Age Uy + 
where the coefficients (A) are to be calculated so as to make v agree with the 
initial state when ¢ = 0. 
Without doing this, however, it is obvious that the proposed explanation 
given above depends for its validity on the supposition that m, is not enorm- 
ously greater than m,; for, if it be, the abnormal terms due to the original 
uniform heating will disappear with very great rapidity. 
A rough calculation showed me that a for the iron bar lies between 2000 
and 3000. Hence the bar is barely out of the bath before these abnormal terms 
have become insensible. The effect due to the finite length of the bar is easily 
calculated by the help of Fourter’s method for a cube, which applies to a 
rectangular parallelepiped of any dimensions, symmetrically heated. It depends 
on the fact that the temperature at any point can be expressed as the product 
of three functions, each containing the time and one only of the coordinates. 
IT owe this hint to Professor CurysTAt. 
Calling 2a, 26, 2c the edges of the parallelepiped, this method leads to the 
following expression— 
sin Na COS na sin nb cos n'y - sin 2”¢ cos n”z A 
v= 64 v3 (Secs bra Qna€ mt), (Sy ae + sin 2n/b © is ‘). > (s + sin 2n’c oe » 
where the values of n, 7’, n” are the roots of 

h hb 
na tan na = - , wb tan n’b = “- nt Guan C= i : 
and v, is the initial uniform temperature. 
With the data contained in the present paper, it is easy to obtain from the 
: id 7 t onren 
above the following values of ae corresponding to a uniform initial tempera- 
ture (v,) of about 200° C., the bar being 14 inches square, by 20 inches in a 
and only the slower vanishing terms te retained :— 
Tron, . = 0:0235¢°%* (1 —0:068e-™) 
Copper (Crown), . —0:0262e~%** (1 —0-06e7?™). 
Hence the rate of cooling is diminished initially as regards the longitudinal 
VOL. XXVUI. PART III. 9E 
