MATHEMATICAL THEORY OF STREAM-LINES. 
279 
In the second expression for the function, 6 denotes the angle made by the trace of 
the cone with the axis of x. 
To draw a set of those traces, describe a circle about the focus ; divide the diameter 
of that circle which lies along the axis of x into a convenient number of equal parts ; 
through the points of division of the diameter draw ordinates perpendicular to it, cut- 
ting the circumference ; through the points of division of the circumference draw radii ; 
these will be the required traces of the cones. 
For the focus of convergence, let x= +«, and for the focus of divergence, let x=—a ; 
then the following function represents the lines of disturbance, or stream-lines of the 
combined motions of divergence and convergence, 
k 2 j x — a 
2[ V{{x— af + if)} 
x + a ] k 2 , 
\Z{{x + a) 2 + y 2 }j = 2 ( C0S 
cos O ') ; 
(32) 
in the last of which expressions 6 and 6' denote the angles made with the axis of x by 
the two lines drawn from the point (x, y) to the foci of convergence and divergence re- 
spectively. Those lines of disturbance are constructed graphically by drawing two equal 
and similar sets of radiating straight lines through the foci, as already described, and 
then drawing curves through the foci, and diagonally through the angles of the network 
made by the two sets of radiating straight lines. Those curves are already well-known, 
being the lines of force of a magnet whose poles are at the foci. The fine curves in 
fig. 3, which spread from the focus A, are examples of them ; they were drawn by the 
method above described, though the radiating straight lines have been omitted from 
the Plate to prevent confusion. 
The stream-lines which are the traces, on the plane of xy, of the stream-line surfaces 
of revolution, may be constructed, as before, by drawing them diagonally through the 
angles of the network made by the parallel straight lines in fig. 3 with the lines of dis- 
turbance. Their general equation is as follows : 
y 8 Ff x-a x + a } 
^—2^2 \\/{{x-a) 2 + y 2 ) */{(tf + a) 8 + «/ 8 }J — \ 66 ) 
b having a series of values in arithmetical progression. The principal properties of those 
lines have been stated in the Philosophical Magazine for October 1864; but their de- 
tailed investigation has not hitherto been published. 
In the particular case &=0, equation (33) has two roots, viz. 
y— 0, representing the axis O X ; and } 
f . .... I • (34) 
p= cos 0 — cos 6, representing the oval of which L B in fig. 3 is a quadrant. 
That oval is the trace of the surface of a solid of revolution which will disturb a 
uniform current in such a way as to produce the whole series of stream-line surfaces 
whose traces are expressed by equation (33); and that oval surface of revolution is 
the only surface of the series which is closed and finite — all the others being inde- 
