410 
ME. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
To reduce the elliptic integrals, we have 
rua ^(,,3 _ 1 ) 
v/(l - «V) 
rVa 1 
dr 
1 v^d - y n. . _ I-' 
= r . . . 
J1 \/ {d — 1 • 1 — a~d) 
rp = 1 
Now (Cayley, ‘ Elliptic Functions,’ p. 315) 
dr dx 
^(,.2 _ 1 . 1 _ adPp v^(l -X- ■ \ - /:V“) 
Thus 
where ^^ and = 
1 - 
L (a) = f 
1 
— ] 
dx 
_1 - (1 - J ^(1 _ 3;2 . 1 _ 
= Oi (I, cfi - 1) - F, {k), 
where TI and F are the third and first elliptic integrals. And 
M(a)= f ’I dp = f 
^ x/(l - rh 
^ .Cy(l_aV) 
dr 
= ^ 1 1 
\ «7 Jo\/(l —r- • l—a?r") «’Jo v/ (1 ■ 
= (l - h F. («) + J; E. (a). 
where E is the second elliptic integral. 
Suppose now the length of the pipe is small, so that a is nearly unity. 
We have 
r x/a-f) 
EV(/“ -«") 
dp. 
If we put p' = X, 2pdp =■ dx, we may put the factor p in the integral 
unity, and so get 
i«V(» — «") 
= ^77 (1 - a^). 
L («) = i I 
J ( 
So that the length of pipe becomes 
/ = (- + 1 j (1 — cd) to the first order ; 
1 
(1 - aV’ 
a-d) 
dY 
equal to 
