424 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
The equation (6) may be written 
(c — a) - = {cm — 1) -t- — 1 ) y(c" — 1) . 
'^2 
Now 
(c — aY' ~ {ac — 1)® — {a^ — 1) (c^ — 1) , 
therefore 
(c - a) = {ac - 1) - ^{a- - 1) ^{c-- 1); 
’’■i 
and, by addition and subtraction, 
From (1) 
Therefore 
or, 
c (^1 ”■ (^3^ - i ('^1 - i ; 
SO that 
_ _ [(^'] - ^'3 + (ly - 
- 1 + G [- - V^) V.^ + (V.J^ - Vi'l'o)]" 
_ (^'3 - ^’ 2 )^ (^’3 + rd' 
K + ^ 3 )® («3 - A)“ ’ 
And 
ac — 1 _ j I «b'\ ^2 
a — 1) ^ \r3 rg/ ?’i — tq ’ 
Also 
{c — a)— c — h from (2), 
therefore 
^’2«3 (^’3^ - 
j_ ( ^ 1 ^ + ^1^0) (rg^ - r\v^ - Ai - rgi^rg- 
2 / “^ \ 5 
(^3 - ^’i^b) 
h = h 
V,,~ + l\ 
v„v. 
_ 1 
2 ' 
('^1 
3''3 
