[SULLIVAN] PARTIAL DIFFERENTIAL EQUATIONS 133 
the associated surface S must be an integral of the equation 
tae a, Sad 
fi 52) 2N 
1.e., the equation 
(r +) (t+ A) = 5? + \'. 
+=", (+=, 
1.€., Z1 =s+% x? + y?), 
and therefore 
Now put 
S1 = S. 
Hence the associated surface S must be an integral of the Monge 
i 1 : : ‘ 
equation s’+rt = av (on changing the notation slightly). 
In addition to effecting the direct integration of the Normal 
System, we have now established the following results: 
The integral surfaces of the Normal System are also integral surfaces 
of the Monge equation (B). 
These surfaces constitute an important subclass of the surfaces whose 
asymptotic curves belong to linear complexes; and are characterized by 
the metrical property that the linear complexes which contain their 
asymptotic curves have their axes parallel to a fixed line (the z-axis), 
and also have their principal parameters constant. 
A metrical characterization apparently entirely unrelated to the 
preceding can be given to these surfaces, namely; they may be regarded 
as the associated surfaces of a certain line congruence whose medial 
surface is a plane parallel to the (xy) plane. 
McGill University, 
Montreal. 
Sec. ITI, Sig. 5 
