THE HON. J. \V. STRUTT ON THE THEORY OF RESONANCE. 
107 
The first expression is obtained by supposing infinitely thin but perfectly conducting 
planes perpendicular to the axis to be introduced from the ends of the tube inwards, 
while in the second the conducting planes in the electical interpretation are replaced by 
pistons in the hydrodynamical analogue. For example, let the tube be part of a cone 
of semivertical angle d. 
The lower limit is 
~ r 2 ) ^(rj+rJ’ 
and the higher 
1 + tan 2 9 / 1 .1\ J_\ 
7r tan S yRj ff4 2 / yRj R 2 J 
Tubes nearly straight and cylindrical but not necessarily of revolution. 
Taking the axis of x in the direction of the length, we readily obtain by the same 
process as before a lower limit to the resistance 
where a denotes the section of the tube by a plane perpendicular to the axis at the point 
x, an expression which has long been known and is sometimes given as rigorous. The 
conductor (for I am now referring to the electrical interpretation) is conceived to be 
divided into elementary slices by planes perpendicular to the axis, and the resistance of 
any slice is calculated as if its faces were at constant potentials, which is of course not 
the case. In fact it is meaningless to talk of the resistance of a limited solid at all, 
unless with the understanding that certain parts of its surface are at constant potentials, 
while other parts are bounded by non-conductors. Thus, when the resistance of a 
cube is spoken of, it is tacitly assumed that two of the opposite faces are at constant 
potentials, and that the other four faces permit no escape of electricity across them. In 
some cases of unlimited conductors, for instance one we have already contemplated — an 
infinite solid almost divided into two separate parts by an infinite insulating plane with 
a hole in it — it is allowable to speak of the resistance without specifying what particular 
surfaces are regarded as equipotential ; for at a sufficient distance from the opening on 
either side the potential is constant, and any surface no part of which approaches the 
opening is approximately equipotential. After this explanation of the exact significance 
of (53), we may advantageously modify it into a form convenient for practical use. 
The section of the tube at n different points of its length l is obtained by observing 
the length X of a mercury thread which is caused to traverse the tube. Replacing the 
integration by a summation denoted by the symbol 2, we arrive at the formula 
resistance = -in- 2 X 2 ......... (54) 
ft ' 2 V v v J 
which was used by Dr. Matte iessen in his investigation of the mercury unit of electrical 
