122 THE ROYAL SOCIETY OF CANADA 
(B) = - = ce I= § = (A=a constant) 
which occurs so frequently in classical works on metrical geometry. 
Thirdly, to extend our knowledge concerning the integral surfaces 
of the Normal System by showing that they can be completely 
characterized by means of certain metrical properties of the linear 
complexes which contain their asymptotic curves; and that they may 
also be considered as the associated surfaces of a certain line congru- 
ence whose medial surface is a plane. 

The integration of the Normal System. 
It is first necessary to show that the Normal System is completely 
integrable. If, for brevity, we designate the coefficients of (A) by 
(2a, 2b; 2a’, 2b’) the system of equations proposed becomes 
0°60 00 00 
/ == 
(A’) CE +2a ae + 2b ae —0, 
dat ape D à 
M se Pc oe 
By differentiation we find a these Fa following equations: 
0°0 00 
@) ous mir i +p ou = tps Ov’ 
00 0? 00 00 
Ou?07 À Bnav Ta ou 19s Ov’ 
00 0? 00 00 
007? ne Oudv Th Ou Ts av’ 
0°0 070 00 00 
Ov Fa Ou Ov Ts ou Ts av’ 
where 
db 
Pi 20; psa? 2, Pro Det 
G= —2a, g=4a'b— 2, gs =4bb’ — a : 
a’ ob’ 
r= —20’, Meret eo M r3= Aa'b—2—, 
ou ou 
/ / 
$;= —2a’, s.=4ab’—2 ~, 53=4b"—2 a 
Now the integrability conditions of (A’) oe oe obtained by 
writing 
fa) 00 0°6 
(3) dv Co) an) 
a [SN à (88 
ov \ ou?0v) du \ dude? J’ 
0/86 \ _ à [20 
dv \ouov° ) au \ av 
