[ ae] 
XIV. On the Singular Solutions of Simultaneous Ordinary Differential Equations 
and the Theory of Congruencies. : 
By A. C. Dixon, M.A., Fellow of Trinity College, Cambridge, Professor of 
Mathematics in Queen’s College, Galway. 
Communicated by J. W. L. GuatsHer, Sc.D., F.RS. 
Received June 7,--Read June 21, 1894. 
INTRODUCTION. 
§ 1. Tuts paper is an attempt to show how the singular solutions of simultaneous 
ordinary differential equations are to be found either from a complete primitive or 
from the differential equations. 
The /atter question has been treated by Maver (‘ Math. Ann.,’ vol. 22, p. 368) 
in a somewhat different way, but with the same result. He also gives a reference to 
a paper in Polish by ZasaczKowskK1 (summarized in vol. 9 of the ‘ Jahrbuch der Fort- 
schritte der Mathematik), and to one by SreRReET in vol. 18 of ‘ Liouvitue’s Journal.’ 
The general result is that there may be as many forms of solution as there are 
variables (the differential equations being of the first order, to which they may 
always be reduced). Each form is derived from the one before by the ‘process of 
finding the envelope, and each contains fewer arbitrary constants by one than the 
form from which it is directly derived. 
The general theory is given in {§ 2, 3, and in § 4 it is shown how the singular 
solutions are to be formed from the differential equations themselves. In {§ 5-9 
the theory is connected with that of consecutive solutions belonging to the complete 
primitive. § 10-13 are taken up with geometrical interpretations relating to plane 
curves aud also to curves in space of n + 1 dimensions, n + 1 being the number of 
variables. In §§ 14-16 the case is discussed in which a system of singular solutions 
is included in a former system or in the complete primitive. 
The rest of the paper contains the application of the theory to certain examples. 
The first example (§§ 17-21) is the case of the lines in two osculating planes of a 
twisted curve, and in particular of a twisted cubic. The particular example is given 
by Maver and Serrer. The second (§§ 22-26) is that of the congruency of common 
tangents to two quadric surfaces, and generally (§§ 27--38) of the bitangents to any 
surface. The third (§§ 39-50) is that of the essentially different kind of congruency 
BB 25.8.95 
