MATHEMATICAL THEORY OF STREAM-LINES. 
281 
are the fine curves spreading from A. The lines of disturbance expressed by 4^ + 4 / « 
being those due to the second pair of foci, are the fine curves spreading from A'. Both 
those sets of lines were drawn according to the rules given in sections 5 and 6. The 
lines of resultant disturbance , expressed by the function \p—\p u = 4 / id- 4 / 2 H- 4 ' 3 + 4 / 45 are 
drawn diagonally through the angles of the network made by the two former sets of 
lines. They are marked with strong dots in the figures. They all traverse one or other 
of the foci, A, A', with the exception of one line, which meets the axis O X at right 
angles in the point M *. 
The actual stream-lines are drawn diagonally through the network made by the lines 
of uniform current and the lines of disturbance. They are shown by rather strong lines 
in figs 2 & 3. In each set of quadrifocal stream-lines there is one only that is finite 
and closed. It corresponds to the value ^ = b= 0; and it is the trace of the surface of 
the solid whose disturbing action produces the whole system of stream-lines. It has 
rounded ends, cutting the axis of x at right angles. In each of the figures 2 & 3, a 
quadrant of that curve is shown, marked L' If. This is the curve which resembles the 
water-line of Mr. Frotjde’s model B, fig. 1, and is therefore properly a cycno'id , or swan- 
like curve. The equation 4 / = 0 has another root, viz. y= 0, representing the axis of x. 
The other stream-lines of the system, lying outside the curve L' B', are infinite, and have 
for asymptotes the stream -lines of the uniform current. They may be called cycnogenous 
stream-lines, as being produced by the cycno’id stream-line surface. 
In a system of bifocal stream-lines there are two independent constants, on which the 
dimensions and figures of all the lines of the system depend — the excentricity (being 
half the distance between the foci) and the parameter (as to which see equation 36). 
In a system of quadrifocal stream-lines, there are five independent constants, viz. : — the 
two parameters, for the first and second pair of foci respectively ; the eccentricity of the 
first pair of foci; and the distances of the two foci forming the second pair from a point 
midway between the first pair. If those distances are equal, the cycnoid curve and each 
of the stream-lines produced by it have then two ends symmetrical to each other ; if 
unequal, those ends are unsymmetrical. In all the examples shown in the Plate the 
ends are symmetrical. 
In each of the figures 2 and 3, the bifocal oval stream-line marked B L has been de- 
scribed about the first two foci with the same parameter which is assigned to those foci 
in describing the quadrifocal closed stream-line B' L'. 
Fig. 4 shows a series of cycnoids, or quadrifocal closed stream-lines, in two dimen- 
sions, described about the same four foci. The parameter for the first pair of foci (one 
of which is marked A) is constant, and is that of the bifocal oval neoid B L. The para- 
meter for the second pair of foci (one of which is marked A') was made successively 
* In fig. 2 the quadrifocal stream-lines and their lines of disturbance have heen engraved on a plate already 
covered with bifocal stream-lines and their lines of disturbance ; and therefore, in order to avoid confusion, some 
of the quadrifocal lines of resultant disturbance extending from A towards the axis of Y, in the neighbourhood 
of the point B', have been omitted. Enough have been drawn to show the principle of their construction. In 
fig. 3 the series of quadrifocal lines of disturbance is complete. 
