THE HON. J. W. STEUTT ON THE THEOEY OE EESONANCE. 
103 
Thus if for brevity we put R=l, we may express the vis viva of the whole motion (both 
extremities included) by 
o • • 7/1 i i \2 i 7 r f *' 2 t — s -8 ' 16/,, 14 5 \ 
2 vis viva=Tl( 1 +f^) 2 + ^ 1 + e - 8 i +y ( 1 + 15 ^ + 2 I f* 
which corresponds to the rate of flow 5ra 0 = sr(l 
Thus 
1 j , ‘ 2 + v( 1 + il ft+ li'‘ s ) 
2 vis viva 
l 
(rate of flow) 2 3ir 
(1 + ^) 2 
(46) 
l_g-8/ 
where A=, , 
1 + e 81 
The second fraction on The right of (46) is next to be made a minimum by variation 
of [jj. Putting it equal to 2 and multiplying up, we get the following quadratic in p : — 
T6-7 z' 
i A 
516 
L 
i 8 
1 21 w 
16 
The smallest value of z consistent with a real value of p is therefore given by 
/16-7 *\ 2 /16 \ /A 5-16 z\ _ 
(isf-s) -(v-d(8 + 2br-4)=° 
2A + 
Z — ■ 
8192 
1575ir_ 2A+ 1-6556 __3'6556- -3444e- 8i 
Ay 12 _ -3927A + *3429” 7356'— 0498£" 8i ’ 
8 '35 
Thus 
2 vis viva l , 1 3-6556 — -3444e~ 8 
Z 1 — -OAQSSc-8/ 
(rate of flow) 2 7r”3ir ’7356 — -0498s" 
This gives an upper limit to -. In terms of a (including both ends) 
(47) 
a < 2-305R. 
IQ'615 — s~ 8 R 
14771 -s _ 4 
(47') 
From (47') we see that the limit for a is smallest when 1= 0, and gradually increases 
with l. 
Now let us cut off a strip of breadth da from the edge of the disk, whose mass is accordingly 
27r«(l + fjta 2 ')da. 
The work done in carrying this strip off to infinity is 
27ra<A<(l + pa 2 )(4a + ^pct 3 )« 
If we gradually pare the disk down to nothing and carry all the parings to infinity, we find for the total 
work by integrating with respect to a from 0 to E, 
8ttRVi , 14 5 A 
nr( + I5'' + 2T'\)’ 
15' 
ju being written for pR 2 . This is, as it should be, the half of the expression in the text.] 
p 2 
