188 Mr J. Brill, On Conjugate Functions [Feb. 27, 
and, therefore, 
dw_1. ais} 
log Go= log {5* * By 
hog (EY + GY} = nn EPR 
= 9 log (=) +) —2 tan lee aar| 
=logh—1. 
If we interpret & as the velocity potential and 7 as the stream 
function of a case of steady, irrotational, two-dimensional, fluid 
motion, then h is the velocity of the fluid at any point, and $3 is 
the angle which the direction of motion at that point makes with 
the positive direction of the axis of a. 
Let w, and w, be two functions of a complex variable, and let 
dW dw, dw, 
(m +n) log ee m log 7 +n log Ee 
™m n 
or aw _ es ae 
dz dz dz 
Then, obviously, W will be a function of a complex variable, and 
may be considered as giving rise to a possible case of fluid motion. 
Let S be the angle made by the direction of motion at any point 
of the fluid in this case with the positive direction of the axis of a; 
also, let S$, and S, denote the corresponding quantities for the cases 
of fluid motion that may be derived from w, and w,; then we 
have 
(m+ n)J=mMsS, +NS,. 
Now suppose that we have the stream lines of these latter two 
cases of motion traced upon a plane, then $, and S, will be the 
angles made with the positive direction of the axis of aw by 
the tangents to two stream lines, one belonging to each system, 
at their point of intersection. Also, 5 is the angle made with 
the same line by the tangent at the said point to the particular 
stream line of the derived case of motion that passes through that 
point. Obviously, this last tangent divides the angle between 
the other two in the ratio »:m. Thus we have the following 
theorem: 
Two families of equipotential curves are traced on a plane. 
A third family of curves is drawn possessing the property that 
the tangent at any point to any member of the family divides in a 
constant ratio the angle between the tangents at that point to the 
particular members of the two former families that intersect in the 
said point. This last family of curves will also form an equipotential 
system, 
