SECTION III, 1916 [121] TRANS: R.SiC: 
Concerning a Certain Completely Integrable System of Partial 
Differential Equations. 
By CHARLES T. SULLIVAN. B.A., M.Sc., Pa.D. 
Presented by JAMES HARKNESS, M.A., F.R.S.C. 
(Read May Meeting, 1916). 
INTRODUCTION. 
The completely integrable system of partial differential equations 
of the second order in two independent variables (uw, v) to be discussed 
in this paper has the form 
dlog @ 00 

(A) D on EU 
0°0 0 aS: pa Me 
== —?— 
Ov? af de a ov? 0, 
where ¢ satisfies the condition lcs: 
COS Oe Ses 
(1) dudv SA 
This system of equations, which we shall call the Normal System, 
arises in connection with many problems in the projective differential 
geometry of curved surfaces; such problems, for example, as those 
investigated by Wilczynski in his paper on Directrix Curves (Math. 
Annalen, Dec. 1914), and problems relating to configurations organic- 
ally connected with surfaces whose Directrix Quadric consists of a 
double plane (Sullivan, C. T.; Trans. Am. Math. Soc., Vol. XV, pp. 
ee Trans. Royal Society of Canada, Vol. IX, Series III, pp. 151- 
16 
These equations have been developed and their integrals 
obtained from entirely different points of view in the papers cited 
above; but the methods of deducing these integrals, though essential 
to a complete analysis of the configurations involved, are somwhat 
circuitous. 
In this paper it is proposed: 
Firstly, to obtain a fundamental set of integrals of the Normal 
System by a direct analytic procedure. 
Secondly, to transform the Normal System in such a way as to 
show that the integral surfaces of this system are also integral sur- 
faces of the Monge equation 
