of determining Stream Lines and Equipotentials. 239 
II. The First Idea of Freehand Solution. 
The real simplicity of the space distributions o£ electric 
and magnetic phenomena, — so much disguised in the algebraic 
integrals of the differential equations, but rescued from con- 
fusion and clearly set forth by the vector analysts, Heaviside, 
Walker and others, — leads one to hope for equally simple 
methods of calculating their numerical values with reference 
to any boundary whatever. 
The beautiful figures of stream and equipotential surfaces 
published by Maxwell, Lamb and others as the result of 
harmonic analysis, and by Hele Shaw as the result of experi- 
ment, suggest that by imitating their characteristic properties 
freehand we may, in some small part, attain the result 
desired. 
Maxwell in § 92 of his 'Elementary Treatise on Elec- 
tricity and Magnetism' speaks of tentative methods of 
altering known solutions of the Laplacian equation by drawing- 
diagrams on paper and selecting the least improbable. The 
object of the present thesis is to point out that this method 
<?an do far more than merely alter known results, and that it 
may be so far from tentative as to yield an accuracy of 
one per cent of the range. 
This method of treating potentials, although still far from 
combining all desirable qualities, and suffering from the 
restriction to certain types of symmetry, yet from its great 
freedom within those types may, it is hoped, supply to a 
certain extent the demand we have indicated. 
On turning to Maxwell's figures and picking out those in 
which V is independent of z so that we have 
yv .w_ n 
it will be seen that while the curves are of the most various 
shapes yet the chequerwork of all the diagrams possesses 
these two properties in common : — (1) the corners are 
orthogonal, (2) when the chequers are small enough the 
ratio of their length to breadth is the same in all parts of 
the field. 
The proof of this follows most conveniently from the con- 
sideration of the motion of a liquid when the lines of flow 
lie in parallel planes and the motion is the same at all points 
of any normal to these planes. Draw three adjacent stream 
lines defining two adjacent tubes of flow. 
S2 
