418 on monge’s “mémoire sur la [782 
of a plane (but, of course, any congruence whatever of lines can be so represented) ; 
and he establishes the theorem that each line of the system is intersected by each 
of two consecutive lines,—viz. taking (xy) as the coordinates of the point of 
intersection of any line with the plane of xy, he obtains, as the condition of inter 
section with the consecutive line a quadric equation in (dx, dy'). He then considers 
the normals of a surface, (which, as lines drawn according to a given law from any 
point of a surface, require a slightly different analytical investigation), establishes for 
them the like theorem, and shows moreover that the two directions of passage on 
the surface to a consecutive point are at right angles to each other; or, what is the 
same thing, that in the two sets of developable surfaces formed by the intersecting 
normals, each surface of the one set intersects each surface of the other set in a 
straight line, and at right angles. He speaks expressly of the lines of greatest and 
least curvature, and generally establishes the whole theory of the curvature of surfaces 
in a very complete and satisfactory manner; the particular case of surfaces of the 
second order is not considered. It may be remarked that, although not explicitly 
stating it, he must have seen that a congruence of lines is not, in general, a system 
of normals of a surface (that is, the lines of a congruence cannot be, in general, cut 
at right angles by any surface) ; he, in fact, assumes (quite correctly, but a proof 
should have been given) that a congruence of lines for which the two sets of 
developable surfaces intersect at right angles is a system of normals of a surface. 
Keverting to the before-mentioned problem (plane or solid), I remark that this 
is a problem of minimum sui generis. Considering the first area or volume as divided 
in any manner into infinitesimal elements, we have to divide the second area or 
volume into corresponding equal elements, in such wise that the sum of the products 
of each element of the first area or volume into its distance from the corresponding 
element of the second area or volume may be a minimum ; but, for doing this, we 
have no means of forming the analytical expression of any function which is to be, 
by the formulae of the differential calculus or the calculus of variations, made a 
minimum. 
For the plane problem, Monge obtains the solution by means of the very simple 
consideration that the routes of two elements must not cross each other; in fact, 
imagine an element A transferred to a, and an equal element B transferred to b : 
the lines Aa, Bb must not cross each other, for if they did, drawing the two lines 
a 
Ab and Ba, the sum Aa + Bb would be greater than the sum Ab -f Ba, contrary to 
the condition of the minimum. Imagine the areas intersected by two consecutive lines 
as shown in the figure: the filament between these two lines may be regarded as