300 
MR. .T. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
The general Theory of Transformation. 
Let X, y be two conjugate functions with respect to the two variables <^, xjj, so that 
+ iy =f{(f> + 
and write 
X iy = z X ■— iy = z\ 
(j) txjj = 'll’ <f> — Ixjj =: 'll’'. 
X, y may be regarded as the rectangular coordinate of a point in a plane which we 
shall call the 2 plane, and similarly xfj are the coordinates of a point in the plane. 
Consider the functions 
. ctz dz 
V = log ~ • VN 3 
^ dv) d.w 
W = — i log — 
dz jdz' 
dvf dv/ 
Since they can be written in the form 
w = 
dw ° dvr 
, dz 
log 
df^ 
dio ” did 
they both satisfy Laplace’s equation, and we have 
V + fW = 2log^3 
' ^ dw 
SO that V, W are conjugate functions with respect to x, y or <j), xp. 
The transformations of the present paper will be deduced from the properties of the 
function V, so that its nature must be considered in detail. We have as alternative 
forms of y 
— {(S)V(im- 
If the element of arc of xjj constant in the z plane be given by 
dcp 
then 
ds^ = 
= m+ 
V = — log Ir 
SO that 
