276 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
Transactions for 1864, page 369. The traces of the axes of divergence and convergence 
on the plane of xy are called the foci. The construction of such bifocal stream -lines 
is represented by the finer and fainter network of lines in fig. 2. OX and O Y are the 
axes of coordinates in the plane of projection, which shows a quadrant of each of the 
stream-lines, the other three quadrants being symmetrical to that shown. The equi- 
distant straight lines parallel to O X are the asymptotes, corresponding to values of y=b. 
A is one of the foci ; and the other is situated at an equal distance from O in the con- 
trary direction. The stream-lines of a current in a plane uniform layer diverging from 
or converging towards a focus are straight, and make equal angles with each other ; and 
their equation is 
^^^tan -1 ^— -—b ; (21) 
in which a—O A denotes the distance of the focus from the origin, b is a constant having 
a series of values in arithmetical progression, and k is a constant called the parameter ; 
so that j is an angle having a series of values in arithmetical progression. This para- 
meter is to be made positive for convergence, and negative for divergence. 
If we suppose the diagram extended so as to show both foci, the focus of convergence 
being in the position x=+a, and the focus of divergence in the position x= — a, we 
obtain for the stream-line function representing these motions combined the following 
expression : 
4' 1 -t-\|/ 2 =^(tan -1 
x—a 
tan 
, x + a \ _ 
V / 
b. 
• • ( 22 ) 
The stream-lines or lines of disturbance represented by this function are constructed 
by drawing two similar sets of equiangular radiating straight lines through the two foci, 
and then drawing curves diagonally through their intersections and through the foci ; 
but as these curves are all circles traversing the foci, it is easier to draw those circles at 
once, without previously drawing the radiating straight lines ; and such is the process 
described in the paper referred to. The fine arcs which traverse the focus A in fig. 2 
are parts of such circular lines of disturbance. Their centres are all in the axis of y ; 
and the radius of any one of them is given by the following formula : let 
— ^=tan 1 X Jr<l — tan" 1 
k y 
x—a 
y 
then radius of circle=« cosec 9 . . 
(23) 
The combination of the divergence and convergence with the uniform current gives, 
for the stream-lines, the comparatively fine curves in fig. 2, which traverse diagonally 
the network made by the parallel straight lines and the fine circular lines of disturbance 
that spread from the focus A. The general equation of those stream-lines is 
4/=3/-|-&^tan' 
x — a 
■tan' 
x-\-a 
y 
)=y- 
■kQ=b. 
(24) 
