THE HON. J. W. STRUTT ON THE THEORY OE RESONANCE. 
105 
da+Ty 2 , if dx be the distance between the planes, the resistance of the unit cube 
being unity. Thus resistance 
= fA* (49) 
J *T 
Upper Limit. 
Secondly, we know that in the case of a liquid the true vis viva is less than that of 
any other motion which satisfies the boundary conditions and the equation of conti- 
nuity. Now u, v being the axial and transverse velocities, it will always be possible so 
to determine v as to satisfy the conditions if we assume u= constant over the section, 
and therefore 
01 . 
(50) 
u= 
This may be seen by imagining rigid pistons introduced perpendicular to the axis. To 
determine v it is convenient to use the function \Jq which is related to u and v ac- 
cording to the equations (37), 
( p 
ru— - t -> 
dr 
rv= — -7-- 
dx 
These forms for u and v secure the fulfilment of the continuity equation. Since 
and therefore 
u 0 
U=—, SJ 
V o( x ) 
y 0 
- r =r—< 
dr try 
'2*y* r ’ 
u o r 4-(. 
r 2it dxyy 2 ' 
But since v cannot be infinite on the axis, but must, on the contrary, be zero, 
%f/ o (#) = 0, 
and we have 
u— 
-?/ 2 
n D 
= _«o 
dx \j / c2 
J.7T 
(51) 
From the manner in which these were obtained, they must satisfy the condition of 
* [It is easy to show formally that no error can arise from neglecting the elfect of the curved rim. . Imagine 
the planes at x and x+dx extended, and the curves in which they cut the surface of the conductor projected 
by lines parallel to the axis. In this way a cylinder is formed which contaius the whole surface between x 
and x+dx, and another cylinder which is entirely contained by the surface. The small cylinder may be ob- 
tained by supposing part of the matter not to conduct, and therefore gives too great a resistance. On the other 
hand, the real solid may be obtained from the large cylinder by the same process. The resistance of the slice 
lies accordingly between those of the two cylinders which are themselves equal in the limit. Hence, on the 
whole, the parts neglected vanish compared to those retained.] 
