114 
PROFESSOR HUGH L. C ALLEN DAE ON 
In carrying: out the electrical method, advantage was taken of the increase of 
resistance of the tube with temperature in order to secure a constant temperature 
gradient in the latter part of the flow-tube by suitably adjusting the current, as 
explained below in § 34. The results were not quite as good and consistent as I had 
hoped to obtain, on account of want of uniformity in the platinum-tube employed. 
I therefore thought it best to defer publication till I could find time to repeat the 
observations under better conditions, but the preliminary work was distinctly 
encouraging, and was particularly valuable as an indication of .effects to be expected 
in steady-flow electrical calorimetry. 
(32.) Superheating of the Central Conductor. 
The case of a glass flow-tube with a concentric conductor, which more nearly 
approaches the arrangement actually employed in the present investigation, leads to 
nearly the same differential equation, but the solution is much less simple. In the 
course of designing the experiment, I worked out the complete solution for this case 
also, including the initial state, on similar assumptions of constant viscosity and 
conductivity. But since the conductor cannot be held exactly central in practice, 
and the other theoretical conditions cannot be realized, the method cannot con¬ 
veniently be applied with a central conductor to the measurement of the conductivity 
of liquids. For the purposes of the present investigation, moreover, since we are 
only concerned with the approximate estimation of a small correction, a less elaborate 
calculation will be more appropriate. In order not to overburden the paper with 
purely mathematical difficulties, it will suffice to give the solution of the limiting 
state for the simpler case in which the velocity of flow is assumed to be constant over 
the cross-section of the tube and equal to its mean value. This simplification does 
not materially alter the general character of the solution, and the numerical results 
which it gives for the calorimeters actually employed are within a few parts per cent, 
of those obtained when allowance is made for the variation of the velocity. 
If we integrate the differential equation (l) on the assumption of a constant 
velocity Y = Q/(rp — r 0 2 )', where r, is the radius of the glass-tube, and r 0 the radius 
of the conductor, writing A for Yc6'/2lk, we obtain the solution 
dd/dr = A r - A r~/r .(7), 
0 0 -6 = Arp log, (r(r Q ) - A (r» - r*)/2 .(8), 
in which the constant is determined by the condition that, neglecting external heat- 
loss, the gradient is zero at the surface of the glass. This gives for the difference of 
temperature between the surface of the wire and the surface of the glass, 
0o - 0i = a Y 2 log* (rjrf) - A (rp - ?y)/2 
(9). 
