230 Mr Briil, On Orthogonal Systems [May 21, 
(4) Orthogonal Systems of Curves and of Surfaces. By J. BRILL, 
M.A., St John’s College. : 
1. Suppose that we have an orthogonal system of curves, and 
let & and 7 be the parameters of the constituent families af the 
system. Then, adopting the usual notation, we have 
ey + -x 
(32) + (Gy) = 
BS 1G OS SOS 
Oz 0% Oy oy 
From these equations we obtain 
0& 0& 0& \* =) 2 
se Mae) ae) h, 
Ch Ein Sia ay 5 (BY ee 
oy Ox (ra oy 
Thus we have, either 
0€ On 0& On 
h, 2a =h, 1 By and h, 2 By has 
OS on 0& _, On 
or haa ape te, hae 
These two forms are virtually identical. In developing the general 
theory we will make use of the first form. It will be necessary, 
however, in applying the theory to any particular case, to examine 
carefully which parameter must be chosen for &, and which for 7. 
0& 0& « (G 0 
lbs Ee di + Bi ay} +h, iz doa ay 
Now we have 
hd& + th,dn _ Ox oy 
da +idy da + idy 
\ he = + th, =| (da + idy) 
= dx + idy 
_, at Bee 
h, 2 ie eae ae 
Thus it is evident that e value of the expression 
h,dé + th,dn 
da + idy 
