234 Mr Brill, On Orthogonal Systems [May 21, 
Thus our equation becomes 
ume: 
{h, ae t hea, ae logh,h, 
0 fh 
+ hh, tae (a ) Fe (logh,h,) += =) 5 ay (log, Dy) 
“hh Seth, BELT on th - 
Expanding and reducing this equation, it becomes 
h? th, h, oh,  h, ch, 0h, -h, oh, ch, 
i GE | ete ie Oe OS Yb Ba ah 
lO GING h, oh) 
=? oe) aaa aa 
This may be written in the form 
a fil Ge IN Gln SG FUN a oGlon 3G 
h, 0& (1) Bbae a : 0 0€ ae an On (;) = 0, 
which is Lamé’s equation. 
In a paper presented to the Society last term, I shewed that 
if we express the equation 
0° logh a 0° logh 
Ox" oy? : 
in terms of the elements of the (A, u) system of curves, we obtain 
the modified form which Lamé’s equation takes when applied to 
that system. I have here shewn that if we express the same 
equation in terms of the elements of the (& 7) system, we obtain 
the general form of Lamé’s equation. This would suggest the 
possibility that some of the other equations obtained in my former 
paper had analogues in the general theory of orthogonal curves. 
The work connected with the reduction of those equations when 
transformed into our present system of co-ordinates would, how- 
ever, be very tedious, and it would not be a prior? certain that the 
portion containing & could be eliminated. 
5. Suppose that we have a second orthogonal system of 
curves, a and @ being the parameters of the two families of the | 
system. Also, let 
Oa\* (0a : 
(a) taley =) =F 
and Cal + Ge) =k. 
