I 889 ] 
V. On the Theory of Free Stream Lines. 
By J. H. Michell, Trinity College, Cambridge. 
Communicated by Professor J. J. Thomson, F.R.S. 
Received Jannai’y 3,-—Read January 16, 1890. 
Introduction. 
The attention of mathematicians was first called to the subject of the present paper 
by a memoir of Helmholtz’s in 1868, on “ Discontinuous Fluid Motion.”'" 
In discussing the steady motion of liquids past salient edges of fixed obstacles, it 
is found that the assumption of continuity of the motion leads to negative pressures 
m the liquid. Helmholtz showed, in the paper above-mentioned, that some cases of 
this kind could be solved by assuming a surface of discontinuity, on one side of which 
the liquid is at rest, and he gave a mathematical solution of one case where the motion 
is in two dimensions. 
The next advance in the subject was made by Kirchhoff who, in 1869, in a paper 
entitled “ Zur Theorie freier Fliissigkeitsstrahlen ” in ‘ Crelle’s Journal,’ gave a 
generalization of the method which Helmholtz had used, and obtained thereby the 
solution of three new interesting cases. Subsequently in his ‘Vorlesungen iiber 
mathematische Physik,’ he published another method and worked out the same 
problems by means of it, but gave no new ones. 
Rayleigh in the ‘Philosophical Magazine,’ December, 1876, discussed the solutions 
of Kirchhoff, and gave a drawing of the bounding free stream lines in one case. 
As far as I know, these are the only investigations published on the mathematical 
side respecting a branch of hydrodynamics of great theoretical and practical interest. 
In considering the method of transformation of polygons given independently by 
Schwarz and Christoffel I have been led to a new transformation, which together 
with theirs, gives a general solution of the problem of free non-reentrant stream lines 
with plane rigid boundaries. 
A considerable number of the cases of high interest prove to be of a tolerably 
simple nature, and I have worked out several in detail. 
These problems occupy the first part of the paper. In the second part I have given 
some extensions of the transformation formulae, which are applicable to problems of 
condensers and the form of hollow vortices in certain cases. 
* ‘Berlin Monatshericlite,’ 1868; and ‘ Gesamm. Abhandl.,’ vol. 1. 
28.6.90 
