MATHEMATICS: G. M. GREEN 
589 
{a'{d - v^^ {h - v)v-{- cix] - a [d' - m„ + + /x {c' - ju)] + m«- ^'J dudv (7) 
+ + (^^, + iU'^)]i/z^2 = 0. 
The consideration of these developable s, together with the focal points 
of the lines / and leads to many theorems which are generalizations 
of known theorems concerning conjugate nets. The point-conjugate 
of the surface S with respect to r', i.e., the surface generated by the 
harmonic conjugate of y with respect to the focal points of V , plays an 
important part in the discussion. 
Of more immediate interest is the case in which the parametric net 
consists of the asymptotic curves of 5. The differential equations of 
the surface may then be written^ 
Juu ^^hy,^fy = 0, + 2 a'y, + gy = ^, (8) 
and the differential equations defining the developables of the con- 
gruences r' and r respectively 
[/ + + - + 2^,] du'' + {y, - M J dudv 
- k + + - 2aV + 2a2j = 0, 
[/ + _|_ ^ 2h^] du" + {v^ - M„) dudv 
-[g + M' + M.+ 2aV]^/?;2^0. 
Especially interesting in this connection are the directrix con- 
gruences defined by Wilczynski.^ These are in the relation R, since 
the directrix of the first kind is the line joining the points 
r = yu-^y, ^ = y.-|iy. (11) 
2a 2b 
and the directrix of the second kind is the line joining the point y with 
the point 
The following new geometric characterization of the directrix congru- 
ences may be given: two congruences T and r' which are in the relation 
R to the asymptotics of a surface S, are the directrix congruences of S if 
and only if their developables correspond to the same net on S. 
A number of propositions, which Wilczynski has proved for the direc- 
trix congruences, subsist also for any congruences r and in the rela- 
tion R with respect to the asymptotic net of the surface. 
