160 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We shall utilise these facts to investigate the nature of the p — q 
discriminant of the partial differential equation of the first order. 
Section II. — Geometrical Treatment. 
Dr J. H. M. Wedderburn in his paper “ On the Isoclinal Lines of a 
Differential Equation of the First Order ” ( Proc . R.S.E., vol. xxiv.) made 
use of the idea of isoclinal lines in a plane to investigate the nature of the 
^-discriminant locus of an ordinary differential equation. In what follows 
I propose to extend the conception of isoclinals to space of three dimensions, 
and by means of it, to investigate the nature of the Cauchian Singular 
Solution, especially the case of the onefold of these solutions. 
The differential equation 
<fi(xyzpq) = 0 (16) 
besides defining a twofold family of integral surfaces, also defines a family 
obtained by regarding p and q as arbitrary constants in (16). Such a 
family we call the isoclinal surfaces of (16), and they are given by 
<f>(xyzab) = 0 . ..... (17) 
Any member of (17) may be regarded as generated by moving from 
point to point of contiguous members of the twofold solution of (16), for 
which p = a, q = b. 
Hence there exists, associated with every member of (17), a twofold 
infinity of infinitesimal planes all perpendicular to the direction 
(aj Ja 2 + b 2 +1, bjs/a 2 + b 2 +l, - ljja 2 + b 2 + 1). 
In what follows these shall be referred to as the integral elements of (17). 
This family furnishes an evident method of describing any member of 
the twofold solution of (16). For starting at any arbitrary chosen point 
on any member (a v b ± ) of (17) we draw an integral element, perpendicular 
to the direction (a 1 : 6^ — 1) which in general meets each of an infinity of 
contiguous isoclinals (a 1 + da v b 1 + db 1 ) in an infinitesimal line. Proceeding 
now to draw integral elements perpendicular to the direction (a 1 +da 1 ; 
b l -}-db 1 : — 1) from each of these to the neighbouring isoclinals, we gradually 
trace out an integral surface of (16). Hence by commencing from every 
one of the twofold of points on any given isoclinal, and proceeding as 
shown above from surface to surface, we map out the twofold integrals 
of (16). 
Now the Cauchian Singular Solution of (16) is the same as the Lagran- 
