60 MATHEMATICS: E. J. WILCZYNSKI 
where 
= D^^Q, D''=NO, (5) 
D and being two of the fundamental quantities of S of the second 
order. The third of these quantities, Z>', is equal to zero on account of 
the fact that the curves x = const, and y = const, form a conjugate 
system on S. 
Let us now apply the Laplace transformations to the surface S, by 
putting 
Zi = Zy -\- az, 2_i = 2^ + bz. (6) 
These expressions determine two points Pi and P_i whose loci give rise 
to two further surfaces Si and S-i. The surface Si is the second sheet 
of the focal surface of the congruence formed by the lines which are 
tangent to the curves x = const, on S\ the surface S-i is connected in 
similar fashion with the curves y = const, on S. 
Consider now the line which joins the points Pi and P_i of the 1st 
and —Ist Laplace transformed nets. There is one such line for every 
point P of the original surface S. Consequently, the totality of these 
lines forms a congruence, whose developables we proceed to determine. 
For this purpose, let us give increments, bx and by, to x and y. The 
coordinates of those points of Si and which correspond to the point 
z -f bx -1- Zyby of S, will be 
Zi =^ zi -\- ~ bx -\- —- by, Z-i = 3-1 + — — ox 4- — — dy. 
ox oy ox oy 
Now we find, making use of (1), 
zi = Zy + az, s_i = z^ + bz, 
^ = -bZy+ (a^ - c) z, = z^^ + bz^ + b^z, (7) 
— = Zyy + azy + ayZ, = - aZx-\- [by - c) z. 
The homogenous coordinates of an arbitrar}^ point on the line Zi Z_i 
will be obtained from XZi and iuZ_i, and this expression, on account of 
(7) , assumes the form 
{\[a {a^ — c) bx 4- {ay + q) by\ + m + b^bx + {by — c) by\ ] z 
-f- {\nQy + /X (1 H- bbx - aby) ] z^ \ [l - bbx ^ {a + p) by} Zy (8) 
4- {ixbx + \mby) Zxx- 
In order that such a point may also be on the Hne Fi P_i, i.e., in order 
that these two lines may intersect, (8) must reduce to a linear homo- 
