MATHEMATICS: E. J. WILCZYNSKI 
61 
geneous combination of Zi and Z-i. This will be so, if and only if 
X, }x, 8x, and dy can be determined subject to the two conditions 
\[k + (ay + q - bn- a'' -ap) by] -h m [{K - b^) 8x + k8y] = 0. 
If we eliminate the ratio X : jjl from (9) , we obtain the differential 
equation of the developables of the congruence, viz.: 
kdx'' -\- [ay + q - bn - - ap - m (K - b^)] dx8y - mkdy^ = 0. (10) 
This equation may also be regarded as the differential equation of 
the curves which the developables of the congruence determine on S. 
The asymptotic curves of S are determined by 
Therefore the curves (10) form a conjugate system on S if and only if 
the simultaneous invariant of (10) and (11) is equal to zero, i.e., if and 
only if hD" — mkD = 0. But, according to (5), this reduces to h = k. 
We have, therefore, obtained the following theorem. Consider a net 
of conjugate curves on a non-developable surface S. Let P be any point 
of this net and let Pi and P_i be the corresponding points of the nets ob- 
tained from the given one by the 1st and —1st Laplace transformations. 
Consider further the congruence of all of the lines such as Pi The 
curves on S, which correspond to the developables of this congruence, will 
form a conjugate system, if and only if the original net of conjugate curves 
has equal Laplace-Darboux invariants. 
I wish to add one further remark. Darboux^ has given a geometric 
interpretation of the condition h = k, different from mine, by means 
of a certain conic in the plane of P, Pi, P_i. 
I have found it advisable to introduce two such conies, which coin- 
cide with each other and with that of Darboux in the case h = k and 
only in that case. By using these conies I have been able, quite recently, 
to interpret geometrically the condition which Bianchi expresses by 
saying that a conjugate system is isothermally conjugate. These sys- 
tems have made their appearance in so many problems of differential 
geometry that such a geometrical interpretation seems to me to be a 
matter of very great interest. I shall, however, reserve the details of 
this interpretation for a place, in its appropriate setting, in a paper on 
the general theory of congruences which I hope to present for publi- 
cation before long. 
1 Legans siir la Theorie generale des Surfaces, vol. 4, p. 38. 
\m8y -f ii8x = 0, 
(9) 
DSx^ + D'V = 0- 
(11) 
