346 
[937 
937. 
NOTE ON THE ORTHOTOMIC CURVE OF A SYSTEM OF LINES 
IN A PLANE. 
[From the Messenger of Mathematics, vol. xxn. (1893), pp. 45, 46.] 
Considering in piano a singly infinite system of lines, then to each point of 
the plane there corresponds a line (not in general a unique line), and we can 
therefore express in terms of the coordinates (x, y) of the point the cosine-inclinations 
a, /3 of the line to the axes. The differential equation of the orthotomic curve is 
then adx 4- fidy = 0, and it is a well-known and easily demonstrable theorem that 
adx + /3dy is a complete differential, say it is =dV; the integral equation of the 
orthotomic curve is therefore V = I (adx + ¡3dy), = const., and we see further that V 
If the lines are the normals of the ellipse — + = 1, then, writing the equation 
of the normal at the point X, Y in the form 
j(x-X) = ^(y-Y), =\ 
U/IA/ y 
a -(- X b -t- X 5 
and therefore 
which last equation determines X as a function of x, y. We have a, /3 proportional 
to —, y; or say we have 
whence 
iff 2 (a+\) 2 + (b + \f ] 
1 x 2 y 2