238 Mr. L. F. Richardson on a Freehand Graphic way 
attracted disk electrometer, parallel plates may be made 
practically infinite by his device of the guard-ring. 
But for the purposes of the engineer this is of very limited 
application. If he is to handle partial differential equations 
freely, they must be applicable to bodies of most various shapes, 
such, for example, as the toothed core-plates of dynamos, the 
water surrounding ship shells and screw propellers, the space 
between turbine blades, and a host of other forms, too irregular 
to be readily described. 
Further than this, the method of solution must be easier 
to become skilled in than are the usual methods with harmonic 
functions. Few have time to spend in learning their 
mysteries. And the results must be easy to verify — much 
easier than is the case with a complicated piece of algebra. 
Moreover, the time required to arrive at the desired result 
by analytical methods cannot be foreseen with any certainty. 
It may come out in a morning, it may be unfinished at the 
end of a month. It is no wonder that the practical engineer 
is shy of anything so risky. 
Harmonic functions have, however, one very strong point 
in comparison to the methods put forward in this paper, and 
that is their accuracy. Once we have determined V as an 
infinite series of harmonic functions, it is usually not much 
more labour to obtain an accuracy of 1 in a million than of 
1 in ten. 
Now it is true that in the determination of absolute electric 
standards measurements are made to 1 in 100,000 or to an 
even greater refinement. But for most chemical and physical 
work 1 in 1000 is more like the limit attained. And in any 
new branch of research, two, five, or even ten per cent, are 
very welcome. The root of the matter is that the greatest 
stimulus of scientific discovery are its practical applications. 
And here, in the design of machinery for example, cost rules 
everything, and this can seldom be foreseen as near as 
1 per cent. 
To sum up. The existing methods of solving Laplace's 
equation are susceptible of great accuracy, but they are slow 
and uncertain in time and, most serious of all, they can only 
be applied to very special boundary conditions. There is 
obviously a demand for a method of solving that group of 
partial differential equations — of which we may regard 
Laplace's as the simplest type — which shall, if necessary, 
sacrifice accuracy above 1 per cent., to rapidity, freedom 
from the danger of large blunders, and applicability to more 
various forms of boundary surface. 
