MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
391 
And 
iW = 
dx 
d4> 
lOO- - 
® dx 
+ t 
. dy 
' d^ 
_ 
d(p d(j) 
3= 2id, 
where 6 is the angle the tangent to the carve xJj makes with the axis of x. 
[Mr. Brill has used the function W as a means of transformation (‘ Cambridge Phil, 
Soc. Proc.,’ vol. 6, and ‘ Messenger of Math.,’ August, 1889), and has thus anticipated 
me in one of the general theorems given in the latter part of this paper as I shall 
notice in the proper place. I was not acquainted with his work when I developed the 
method here given.] 
Let k be the curvature of the curve v// at <5?), We have the well-known formula 
which is 
k 
dh 
d'^ 
Now let the arc of the curve ifj be connected with k by the equation 
( 1 ) 
so that 
or from (P 
. =f{k) 
l± 
. . ( 2 ) 
If if/ consist of parts of straight lines we have simply 
— — 0 
d-\^ 
( 3 ) 
The formula (2) suggests a general method for finding a transformation 
- 
such that xJ/q is an arbitrary curve in the z plane. 
If the region within \pQ corresponds, point for point, to the part of the w plane lying 
above xfj = xf/g, the problem is reduced to finding a potential function V, which is 
continuous throughout the space bounded by a straight line, and such that 
over that straight hue 
