THE II OH. J. W. STETJTT OH THE THEORY OF EESOHAHCE. 
79 
which I have not hesitated to avail myself freely both in investigation and statement. 
Much circumlocution is in this way avoided on account of the greater completeness of 
electrical phraseology. Passing over the case of mere holes, which has been already 
considered by Helmholtz, and need not be dwelt upon here, we come to the value of 
the resistance for necks in the form of circular cylinders. For the sake of simplicity 
each end is supposed to be in an infinite plane. In this form the mathematical problem 
is definite, but has not been solved rigorously. Two limits, however (a higher and a 
lower), are investigated, between which it is proved that the true resistance must lie. 
The lower corresponds to a correction to the length of the tube equal to ^ x (radius) 
for each end. It is a remarkable coincidence that Helmholtz also finds the same 
quantity as an approximate correction to the length of an organ-pipe, although the two 
methods are entirely different and neither of them rigorous. His consists of an exact 
solution of the problem for an approximate cylinder, and mine of an approximate solu- 
tion for a true cylinder ; while both indicate on which side the truth must lie. The 
final result for a cylinder infinitely long is that the correction lies between -785 E and 
•828 E. When the cylinder is finite, the upper limit is rather smaller. In a some- 
what similar manner I have investigated limits for the resistance of a tube of revolution, 
which is shown to lie between 
and 
where y denotes the radius of the tube at any point x along the axis. These formulae 
apply whatever may be in other respects the form of the tube, but are especially valu- 
then very near each other, and either of them gives very approximately the true value. 
The resistance of tubes, which are either not of revolution or are not nearly straight, is 
afterwards approximately determined. The only experimental results bearing on the 
subject of this paper, and available for comparison with theory, that I have met with are 
some arrived at by Soxdhauss * and Wertheim f . Besides those quoted by Helmholtz, I 
have only to mention a series of observations by Sondiiauss J on the pitch of flasks with 
long necks which led him to the empirical formula 
n = 46705 
L 2 S 2 
<t, L being the area and length of the neck, and S the volume of the flask. The corre- 
sponding equation derived from the theory of the present paper is 
Pogg. Ann. vol. lxxxi. 
t Annales de Chimie, vol. xxxi. 
M 2 
± Pogg. Ann. vol. lxxix. 
