1888. ] of Curves and of Surfaces. 235 
Then the values of the two expressions 
h,d& + th,dn kda + tk,dB 
PECTS) an 
are each independent of the ratio dy: dx. Hence, if we suppose 
them both to refer to any, the same, variation made in an arbitrary 
direction from a given point, the value of the expression 
hdé + th,dn 
k,da + tk,dB 
will be independent of the direction of that variation; 7.e. the 
value of the said expression will be independent of the ratio 
da:df. This gives us the two relations 
nk, = hh, 3 sal Tle = = Hye ae 
6. We will now attempt to develope with respect to ortho- 
gonal systems of curves traced upon any surface whatsoever, a 
theory analogous to that we have developed with reference to 
orthogonal systems of curves traced upon a plane. Let & and 7 
be the parameters of the two families belonging to one orthogonal 
system, and « and 8 those of the two families belonging to another 
orthogonal system. The length of an elementary arc will be given 
by a formula of the type 
2 _ de 
ds ~ he a2 he 
in the first system of co-ordinates, and by’a formula of the type 
ds Zi dae , a8" 
ae ae ; 
in the second system of co- sonic Hence if we express any 
given elementary are in terms of the elements of both systems 
and equate the results, we have the Si 
0& 0& on 4 \ eas Be 
re lan t+ 5 2} + tga tg Oh mt Ee 
This equation holds aan of the value of the ratio 
da:df8. Thus we have 
n 1 
(gz) = sete 
he v3 (ge) ei 
ia (ae) tae (Ga) ~ ES 
i Occe ak CMGHas 
he On OB he On 0B 
VOL. VI. PT. IV, 17 
