dy 1 1 1 therefore = {e (aa) + ( a − a)3). dx 2 1 1 n 1 (a)) + с 1+ n 8. The class of problems which we shall next consider is introduced chiefly on account of the instructive light which it throws upon the singular solutions of differential equations of the second order. Inverse Problems in Geometry and Optics. The problems we are about to discuss are the following: 1st, To determine the involute of a plane curve. 2ndly, To determine the form of the reflecting curve which will produce a given caustic, the incident rays being supposed parallel. In both these problems we shall have occasion in a particular part of the process to solve a differential equation of the first order of the form y - ad (p) = ff 1 (p) - (p) f1 $ (p) ...... (7), in which and fare functional symbols of given interpretation, and f is a functional symbol whose interpretation is inverse to that of the symbol f'. Thus, if ƒ (x) = sin x, then f'(x) = cos x, ƒ'1(x) = cos ̄1x. It will somewhat less interrupt the theoretical observations for the sake of which the above problems are chiefly valuable, if we solve the equation (7) under its general form first. Referring to Chap. VII. Art. 7, we see that (7) will become linear if we transform it so as to make either of the primitive variables the dependent variable, and either p or any function of p the independent variable, Let us then assume $ (p) = v, and transform the differential equation so as to make x and v the new variables. whence x = e ̄¥v) {C′ + fe¥•) 4' (v) ƒ'~' (v) dv} = e ̄¥v) {C' + e¥o) ƒ'−1 (v) − fe¥o) dƒ'−1 (v)}, x − ƒ'−1 (v) = e−4(v) | between which and (8), v must be eliminated. If in those equations we make f(v) = t, they assume the somewhat more convenient form, and these may yet further be reduced to the form From these equations it only remains to forms of ƒ and 4 being specified, and that of and this is apparently the simplest form of the solution. eliminate t, the given by (9); 9. We shall now proceed to the special problems under consideration. To determine the involute of a plane curve. It is evident from the equations which present themselves in the investigation of the radius of curvature, that if x, y be the co-ordinates of any point in a plane curve, and x', y' those of the corresponding point in the evolute, then Art. 320). Hence, if the equation of the evolute be y' = f (x') we shall have on substituting therein for y and x' the values above given, .(12), y, a differential equation of the second order connecting x and and therefore true for each point of the curve whose evolute is given. Of that evolute the curve in question is an involute. Hence, if y=f(x) be the equation of a given curve, the equation of its involute will satisfy the differential equation (13). Now suppose that nothing was known of the genesis of the above equation, and that it was required to deduce its complete primitive, and its singular solution, should such exist. Upon examination the equation (13) will prove to be of a kind analogous to that of Chap. VII. Art. 9. If we assume a and b being arbitrary constants, we shall find that each of these leads by differentiation to the same differential equation of the third order, viz. 3pq-(1+p2) r = 0... (16), d3y where r stands for d. It follows hence, that a first integral of (13) will be found by eliminating q between (14) and (15), and connecting the arbitrary constants b and a by the relation b=f(a). Eliminating q, we find x-a+(y-b) p=0 wherein making bf (a), we have x-a+{y-ƒ(a)} .(17), (18), for the first integral in question. Again, integrating, we have (x − a)2 + {y — ƒ (a)}2 = p2. .(19), in which a and rare arbitrary constants. This is the complete primitive of (13). It is manifest from its form that it represents, not the involute of the given curve, but the circles of curvature of that involute. Indeed, that the complete primitive cannot represent the involute might have been affirmed a priori. The equation of the involute of a given curve cannot involve in its expression more than one arbitrary constant; for the only element left arbitrary in the mechanical genesis of the involute is the length of a string. It remains to examine the singular solution of (13). This is most easily deduced by eliminating a between the first integral (18) and its derived equation with respect to a, viz. between the equations x−a+{y-ƒ(a)} p=0............(20), M |