104 



(r' — r).dx' + 2(s'^s).dx.dy+(if — t).dy^=0, 

 that is 



dp', dx + d^. dy = dp.dx + dq. dy. 



Adding therefore the two equations (Z), multiplied respectively by {dx, dy), then changing 

 {dp'dx + dijfdy) to [dpdx + dqdy), and reducing; we find the following general expression for 

 the osculating focal distance 



d^-^df + dz^-de 



* (y + y)' ( dp.dx 4- dqdy) + rf«'.rfx + d^'.dy + rfy'.rfz * ^ ' 



To simplify this formula, let us take the given reflected ray for the axis of (z) ; the numerator 

 then reduces itself to {dx"^ -}- dy'^), and the denominator may be put under the form 



tdx* ■\-t,dxdy + %dy*, 



the coefficients i, ^, ti, being independent of 5, and of the differentials ; if then we put dy = dx. 

 tan. ^, so that (if') shall be the angle which the plane, passing through the ray and through the 

 direction of osculation, makes with the plane of (xz), we shall have 



— = j. cos. '4" + ?• sin. 4'' COS. ■<p +». sin. '■^. (W) 



This formula may be still further simplified, by taking for the planes of (x, z), {y, z), the tan- 

 gent planes to the developable pencils, which, by what we have proved, correspond to the maxi- 



mum and minimum of (g ) . To find these planes we are to put -77- ss 0, which gives, tan. 

 2.4, = —3 — ; if then we take them for the planes of (x, 2), (y, 2) we shall have 



«i ' ea ' 



and the formula for the osculating focal distance becomes 



— = — . cos. 'il' + — . sin. ^4^. (C) 



5i> ?2> being the extreme values of 5, namely the distances of the two points in which the ray 

 touches the two caustic surfaces. The analogy of this formula (C') to the known formula for 

 the radius of an osculating sphere, is evident ; and it is important to observe, that although the 

 reciprocal of (j) is included between two given limits, the quantity {{) itself is not always in- 

 cluded between the corresponding limits, but is on the contrary excluded from between them, 

 when those limits are of opposite algebraic signs, that is, when the two foci of the ray are at op- 

 posite sides of the mirror : so that, in this case, there is some impropriety in the term greatest 

 osculating focal distance, since there are some directions of osculation for which that distance is 

 infinite, namely, the two directions determined by the condition 



