236 Mr Brill, On the Orthogonal System [May 21, 
From these equations we obtain 
IL Gls 1 On 1 =) 4 ap al 
h, 0a h, 0a a lz ale 
ay FD i Oe ons 
h, 0B h, 08 bane 0f/ h, \08 k 
If we adopt the upper ne we have the two relations 
0& Oe On 
hk, ag a ap and hk, = a8 = —hk, aa 
These relations enable us to write the above equations in a 
different form, viz. 
hb? (ey + he (=) = he, 
Gay san a) =a 
2208 On ee On 
1 Oa aan 2 08 08 
= ()) 
Now we have 
h,d& + th,dm =i, ire da + YG 53 | +h Ay da + a8 =n al 
0€& h,k, On On hike, 0& 
= |), 2 Ba eS a, OB + th, sae ee am de 
=? cE (la-+ tkydB) + ie 5 = (I da + tk) 
a aie oF Ah, =| (k,da + 1k,dB). 
Therefore, we have 
hdé+thdn 1 {, 0,» On) _ an, a€) 
k,da+ik,dp k, \h, a" pal “eal ae Maeeae 
7. The fact that the equations connected with our present 
theory are identical in form with those connected with the most 
general theory for the plane, as developed in Art. 5, would lead 
us to expect that the simpler forms of the plane theory also had 
their exact analogues in the theory relating to surfaces. That we 
may develope a theory exactly analogous to the theory of functions 
of a complex variable as applied to the plane, has been shewn by 
