244 Mr. L. F. -Richardson on a Freehand Graphic way 
error was 0*84 per cent, of the range, a degree of accuracy 
which would be sufficient for many purposes. 
Having shown that the freehand method is a practical one 
in a plane, it will be well next to enquire to what types of 
symmetry it may be extended. 
III. Possible Types of Symmetry. 
Inasmuch as a single chequerwork is to determine the dis- 
tribution in the whole of the space considered, we are confined 
to two coordinates. The freehand method at present offers 
nothing to compete with the analytic forms in which three 
coordinates appear, such as : 
Y = 2 ( A m sin mx + B OT cos mx) (C n sin ny + D cos ny) (Fe~ Jm*+n? . z -f Ge + ^m2+n 2 . . 
Y = X ^ ± ^(A w cos m$ + ¥> m sin m<j>)J m (kr), 
m,Jc 
(Whittaker, ' Modern Analysis,' p. 318) 
Y = 2 C m r m (A n cos ncf> + B n sin n£) (sin 0) » ^^*, 
(Byerly, ' Fourier's Series/ p. 196) 
or others like them. 
The expressibility of Y in terms of two coordinates implies 
that Y is constant along a certain family of lines in space, 
namely, the intersections of the surfaces over which the said 
two coordinates are respectively constant. Any particular 
type of symmetry is most conveniently distinguished by 
specifying the family of lines along each of which Y must be 
constant. 
As it will frequently be necessary to refer to these lines 
and to distinguish them from the normals to the surfaces 
Y=constant, I propose to call them the u guiding lines/' 
It is indeed conceivable that by adding together several 
space distributions in each of which Y is constant along a 
different family of lines, we might attempt the solution of 
problems which it may be impossible to treat by two co- 
ordinates directly, such for example as 'the motion of a perfect 
fluid past a three-bladed screw-propeller, or the electrostatic 
field due to a ring of electrons. It may even be possible to 
treat the most general distributions by means of sections of 
the potential surfaces drawn on the leaves of a book of tracing- 
paper. But these extensions must be left to those who desire 
the results. This paper deals only with two coordinates. 
