the calculus of functions. 
253 
Problem XLII. 
Required the nature of a curve such that taking any ordi 
nate 
DB, and drawing a normal at the point D, the next ordinate 
CE raised at the foot of the normal shall be equal to that 
normal. 
Let AB = x, BD =y and y = be the equation of the 
curve, then BC == 
and DC = \/ y * -f J^j'and by the condition of the Problem 
we have 
hence 
cw + 
~'\sxd^x~\ '■ 
dx J 
This is apparently a very difficult functional equation, and I 
am not acquainted with any direct method of solving other 
similar ones. It is in fact only from a peculiar condition which 
this equation involves that any solutions have been obtained. 
the condition to which I allude is, that the quantity 
J'* d-\fX ! 
• • , ■ ■ ■■ does 
not change, when for x we substitute x + or expressed 
in symbols, that 
LI 2 
