THE HON. J. W. STEHTT ON THE THEORY OE RESONANCE. 
101 
The second term vanishes in virtue of (36), and we may write 
Twice vis viva=ulrl-\-{ (Ay 1 -\-¥>y' 2 )dx, 
(40') 
where A and B are known quantities depending on £, and y=<p(x) is so far an arbitrary 
function, which we shall determine so as to make the vis viva a minimum. 
By the method of variations 
y= Cr^’+C'g+^r; (41) 
and in order to satisfy (39), 
1=C + C, 
l=Ce-^£ l +Ce + ^i l j 
(42) 
(41) and (42) completely determine y as a function of x, and when this value of y is used 
in (40) the vis viva is less than with any other form of y. On substitution in (40'), 
Twice vis viva =.yfal + 2-v/ AB 1 ~~ g ^ A. . .......... (43) 
The vis viva expressed in (43) is less than any other which can be derived from the 
equation (38); but it is not the least possible, as may be seen by substituting the value 
of in the stream-line equation 
<H> d^ 
dx 2 r dr * dr 2 ’ 
which will be found to be not satisfied. 
The next step is to introduce special forms of Thus let 
u x=0 —l-\-yr. 
Then u 0 = l+if*, 
i+r 2 )- 
Accordingly 
A=^, 
12 
16 12’ 
\/AB=— ; 
v 48 
and (43) becomes 
2 
vis viva =kI(1 -f 
7 TjX 2 1 — <-~ 8t 
24 1+e- 8 ' 
(44) 
We have in (44) the vis viva of a motion within a circular cylinder which satisfies the 
continuity equation, and which makes over the plane ends 
u= l-j-^r 2 . 
If (m = 0 we fall back on the simple case considered before ; and this is the value of ^ for 
which the vis viva in (44) is a minimum compared to the rate of flow (1 But 
for the part outside the cylinder the vis viva is, as we may anticipate, least when (x has 
mdccclxxi. p 
