20
p′=u·2pu√1−p2u2=2p√1−p2u2.
Then if for v be put its value ufcosφu−fcosφ, or dudlp2u, and for p the proper function of u given by the equation to the original curve, and u be then eliminated, we shall have an equation in u′ and p′, which will be that of the caustic.[1]
25. In order to obtain an equation in rectangular co-ordinates, we may proceed as follows:
Reasoning as before, since the caustic is formed by the continual intersections of the reflected rays, two of these are necessary to determine one point of the caustic, and the point where one of them meets the caustic, is that which it has in common with the next; so that if we refer the two reflected rays to the same abscissa, their ordinates, differing in general, coincide at this point, and as far as that point is concerned, a change in the point of the reflecting curve, or in its co-ordinates, takes place without any alteration in the co-ordinates of the reflected ray (Fig. 19.)
We have therefore only to find the equation to the reflected ray belonging to an assumed point of the curve; to differentiate this, considering the co-ordinates of the curve as the only variables, and eliminate these co-ordinates between this equation, its primitive, and that of the curve.
An Example will make this more intelligible.
To diminish the length of the process, we will confine ourselves to the simple case of parallel rays, and take one of them for our
- ↑ For instance, if the reflecting curve be a logarithmic spiral, and Q its pole, its equation is of the form p=mu,
v=du2dpp−duu=u; p′=2mu√1−m2; u′2=4u2−4m2u2·uu=4u2(1−m2),
whence we find p′=mu′; and the caustic is therefore another logarithmic spiral differing from the former only in position.
