MR. J. H, MICHELL ON THE THEORY OF FREE STREAM LINES. 
425 
so 
that 
and 
, j_ (^1^ + ^7l»3 + 2V^V^) - V^V^) - (t>i - ^ 2 )=^^ 
^ V 2 U 3 f^’ 3 ^ — V-^V^) 
1 _ 7 _ 1 (I's^ — + (V-^ — 
^ 2 / ^ \ • 
V 2 '>h (^Y - ^’ 1 ^ 2 ) 
These may be reduced to the form 
1+6 
1 -6 
l(di±+^ 2^3 - 
V, 
2 2 
1^2 
. . 1 (% - 2^3 +^1 + ^2 
2 2 
Vo^ — V-,Vc 
Further, 
a — h = ^ 
— '^d) (V — '^ 1 ^ 2 ) + — '^ 2 )^ 
(^'3 - ^l'^2) 
1 (^3^ - '^ib" 
^ Vl'Ts (V 32 - 
1 _ 1 {•i^3^^1^2 - (V + 'i’i^) (^d + '^^ 1 ^ 2 )} {^3® - '^ 1 ^ 2 } + (^3^ + '^1®) K - '*^ 2 )%® 
i UU - ^ 2 / 0 \ 
^1^2'^3 (^ 3 ” - '^ 1 ^ 2 ) 
_ ^ (ri + ^ 2 ) (^ 3 " - 
’ (V - '^^ 1 ^ 2 ) 
Whence, finally, 
a — h 
c — h 
So that we get for hjl 
TT — 
+ 
log 
^3 
■ ? 
2^3 — 
— '<’2 
2^3 + 
^1 
+ "^2 ' 
+4^3 
0 
■ + '?‘2)^ 
^3 
2^3 
- - ^'2 
■ ^i)’“ 
2^3 
+ ^1+^2 
i% — 
^’2) ('^’3 + t'l) 
^3 (^^'l - ^2) d^’3 + ^2) (% - ^1) 
4f’3^ — (Vl + ^’ 2 !^ ‘ 
+ a TT 
3 I 
MDCCCXC.—A. 
