1888. ] of Curves and of Surfaces. 237 
Beltrami*. We will, however, shew how this theory may be de- 
duced from our present results. 
The value of the expression 
(m + in) (h,dé + th,dn) 
(p + tq) (k,da + tk,dp) 
will be independent of the value of the ratio da: d8; and, writing 
=m-+in and g=p+iq, we will seek to determine f and g so 
that the expressions f(h,d&+th,dn) and g (k,da+ik,dB) may be 
perfect "ee rh necessary ie for this will be 
= (fh) =i (jh,) and = a8 J ip) = iS (gh), 
or ho eeah eth \ge a itl 0 
: " "OE"! lon ° OF 
; Ce (aL 
and ee ae — tk, = A ar) 08 — sa = (0), 
These equations will enable us to determine suitable forms for 7 
and g, and the solution of each will involve an arbitrary function. 
Further, if we write 
dw =f (h,d& + th,dn) and d€=g (k,da+k,dB), 
then dw/d& will possess a’ single definite value; and, if we further 
write w=A+%m and C=y+716, we see that the value of the 
expression 
dn +1dp 
dry +1d6 
is independent of the value of the ratio dy: dé. This necessitates 
the relations 
Thus we see that X+i~=F (y+ 120); and, consequently, we are 
furnished with a theory exactly analogous to the theory of functions 
of a complex variable as applied to the plane. 
Separating the equation 
(p+1q) (k,da + 1k, dB) = dy + dd 
into its real and imaginary parts, we have 
pk,da— qk,dB=dy and qk,da+pk,dB = dé. 
* “Delle variabili complesse sopra una superficie qualunque”—Annali di Mate- 
matica (2) 1., 329—366. The essential elements of the theory are, however, due 
to Gauss. 
17—2 
