24:8 Mr. L. F. Richardson on a Freehand Graphic way 
whence by (Lame, § xxx. 21) 
Br(^l) =0 ' MVjr - 
But if the surfaces 7 are planes, then by the equations 
(Lame, § xxx. 21) already quoted 
A(L\ =0== A(L) 
"dy\ha/ ~dy\hfiJ 
and consequently ~— y and — ~ are independent of 7. But it 
r a r$ 
is shown by Lame (§ xxxviii.) that the curvature of the arc 
of intersection of the surfaces a and {3 is equal to 
So that if the radius of curvature of this arc be p then p is 
independent of 7. But p is equal to the length of the normal 
from the point considered onto the line of ultimate inter- 
section of two consecutive planes of the family 7 which pass 
one on either side of the point considered. As the plane 
moves this length must remain constant. And as this is to be 
true for every point in space, it is easy to see that if the 
surfaces 7 are planes they must intersect in a common axis. 
We have in this case symmetry about an axis. Or if the 
axis be at an infinite distance, the planes are parallel, and we 
have V independent of one of the Cartesian coordinates 
x, if j z. But if, quite generally, the surfaces 7 are spheres 
ha • r7 
we have only -~ independent of 7 and therefore _fL inde- 
\ r l 
pendent of 7. If the centres of the spheres 7 lie in a straight 
line, then since the orthogonal traces of the surfaces « and /5 
on a sphere 7 may turn round anyhow, we may choose for /3 
the planes intersecting in the line of the centres of the 
spheres 7. Then rj[ = 0, and consequently r\ is independent 
of 7 so that the traces of a = const, on the planes /3 are 
circles. This is the system of toroidal coordinates which has 
been treated by Professor Hicks in Phil. Trans. 1881, 
Part II. Now the above reasoning would lead us to expect 
in these a type of symmetry which can be dealt with by two 
coordinates — other than symmetry about an axis. But on 
referring to Hicks's formulae it is easy to show that this is not 
possible for if V be made independent of either of those two 
of his coordinates which determine position in a plane passing 
through the axis, then the other of these two will not divide 
