392 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
We have still to discuss the question of the singular points of V^. 
The function V will be finite and continuous for all points except where two 
branches of a i// curve {or o, (f) curve) cut. 
At such a point dz/div is either zero or infinite, and in either case V is infinite. 
It will be sufficient for our purpose to consider only the simplest singularity, that 
is, in which we have in the neighbourhood of a point ’Ao) ^f this nature 
Y = n log [{(f) — (^o)® + (v// — + C, 
and therefore 
= A (^ “ '^u)b 
where the value of n will depend on the nature of the singularity in question, and 
will be seen from the particular problems to which we proceed. 
Problem I. 
To find the transformation 2 = f{(f) -fi ixjj), which makes the area for which ip is 
positive in the w plane correspond point for point to the area inside a given rectilinear 
polygon in the z plane. (The problem of Schwarz'"' and CHRiSTOFEELt.) 
Consider the conditions which the function V must satisfy in the o) plane. 
(a) There are to be no singular points for xp positive. 
{b) Along i/; = 0 we have dYIdxjj = 0. 
(c) At certain points of \p = 0 , which correspond to the angular points of the 
polygon in the 2 plane, we have V infinite. 
It is plain from this specification that the function V is (to a constant) merely the 
potential of masses at the singular points (pi, cp^ . ■ . along ip = 0 ; and, therefore, 
V = log n{{(p- <p,f -b + C, 
so that 
S = nA («.' - 4,)- 
where 11 is the product symbol. 
It remains to find the quantities n,.. 
the point (pr, 
Draw a small semicircle of radius P around 
* “ Ueber einige Abbildungsaufgaben ” (‘Crelle,’ vol. 70, 1869). 
t “ Sul problema delle temperature stazionarie,” &c. (‘Annali di Matematica,’ vol. 1, 1867). 
