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XXL A Freehand Graphic ivay of determining Stream 
Lines and Equipotentials. By L. F. Richardson *. 
[Plate XII.] 
Scheme oe Paper. 
I. On the need for new methods. 
II. The first idea of freehand solution and confirmation of its 
accuracy. 
III. The conditions which the solution of V 2 V=0 must satisfy in 
order that it may be determinable by a single graph. 
{a) When the guiding lines are normal to a family of surfaces. 
Possible types — test cases. 
{b) Thin shells, 
(c) Screw symmetry — example. 
IV. Points and lines of equilibrium. 
V. Equations other than Laplace's — variable conductivity. 
VI. Boundary conditions. 
VII. Miscellaneous notes on draughtsmanship. 
VIII. Estimation of errors. 
T 
I. The Need for New Methods, 
HE Laplacian differential equation 
B^ 2 V "*" ** 2 " 
has received an extraordinary amount of attention during the 
last century owing to the great number of physical quantities, 
the space distribution of which can be determined from its 
integrals. The analytical integrals hitherto obtained by such 
means as Fourier series, Bessel functions, spherical and other 
harmonics make it possible to determine the distribution when 
the boundary conditions bear relation to certain simple types 
of surface, such as parallelipipeds, cylinders, spheres, ellipsoids, 
anchor rings, &c. 
Now for physical research this is well enough. It is 
usually possible to arrange the instruments so that the parts 
involved are of these simple forms. The wires may be wound 
in circular rings of small cross-section, as in Helmholtz's 
galvanometer. The pieces of substance for the measurement 
of specific properties may be shaped into square bars, as in 
Forbes's experiments on the flow of heat. Or, as in Kelvin's 
* Communicated by the Physical Society ; read November 8, 1907. 
Phil. Mag. S. 6. Vol. 15. No. 86. Feb. 1908. S 
