264 Mr. L. F. Richardson on a Freehand Graphic way 
This being so, the general form of V in all three cases is 
Y = a(u 2 -v 2 ) + b . uv+g(u 3 -3uv 2 ) + h(v*-3u 2 v) + higheY terms, 
where a, b, y, and h are arbitrary constants. When the 
ratios of a and b to the succeeding coefficients do not vanish, 
then the first two terms are all that we need consider. Now 
it may easily be shown that by a proper rotation of the axes 
of reference, so that u v tranf orm to u x v^ the sum of these 
two terms may be transformed into either of them separately. 
We need therefore only consider one, say bu^. The con- 
tours of this function are hyperbolas and are orthogonal to 
those of a(u-f— i^ 2 ). The ratio y is determined by the chequer 
ratio in the neighbourhood of the equilibrium point. 
A graph of this function for the special case of unit 
chequer ratio is given in Webster's ' Dynamics,' p. 525, and 
shows that two equipotentials meet at right angles at the 
equilibrium point, and that two stream-lines also pass through 
the same point and bisect the angles between the equi- 
potentials. The eight curved chequers which meet in the point 
each have consequently three corners of 90° and one of 45°. 
A graph of this function may be used as a " standard equili- 
brium point " to keep the eye informed of the necessary 
proportions of the first and second ring of chequers sur- 
rounding the point. 
If, however, the coefficients a and b vanish, while (j and h 
do not, then the terms of the 3rd degree become all 
important. 
By rotating the axes the sum of the two terms of the 3rd 
degree may be reduced to either separately. A rough graph 
of the contours of these functions is given by Fiske in 
Merriman & Woodward's ' Higher Mathematics,' p. 248. 
Here three equipotentials intersect in the equilibrium point. 
And three stream-lines bisect the angles of 60° which are 
formed in this way. 
Now when a graph has to be drawn and is found to contain 
an equilibrium point, the general arrangement of the potential 
will give us the clue as to whether two, three, or more equi- 
potentials intersect in the equilibrium point. And this being 
known, we have only to draw in the standard type at the 
proper dimensions and chequer ratio. 
When the graph is drawn on a plane passing through an 
axis about which there is screw symmetry of the sort described 
in Section III c, then the appearance is different, for we 
have to add to the value of V 2 V for circular symmetry about 
