108 



whfefl the pencil is given. It follows from this theorem, that along a given ray the density of the 

 reflected light varies inversely as the product of the distances from the two foci, and is infinite at 

 the caustic surfaces. •'^,. „f -it'-,' 



[44.] The same equations (E'), from which we have deduced the theory of thin pencils, serve 

 also to investigate the properties of other undevelopable surfaces, composed by the rays of the 

 system. The most remarkable difference between an undevelopable and a developable pencil, 

 consists in this, that the tangent plane to the latter always touches it in the whole extent of a 

 ray ; whereas in the former, when the point of contact moves along a given ray, the tangent 

 plane changes position, and turns round that ray, like a hinge. To find the law of this rotation 

 let the coordinate planes be chosen as before, the given ray for axis of (2), the point where it 

 meets the mirror for origin, and the tangent planes to the two developable pencils for the planes 

 of (xz), {yz) ; then by (E'), the equations of an infinitely near ray will be 



x'=(«' — 5, )•«?«. ,y=(z' — 5,). rf/3, (K') 



and if it belong to a given undevelopable pencil having for equation ^ =y (a), we shall have 



d/i =/'. d», 



J' being a given quantity ; the tangent plane to this pencil, at any given distance (2') from the 

 mirror, being obliged to contain the given ray, and to pass through a point on the consecutive, 

 has for equation 



f = |Ef-/'= (L'> 



X z — g, 



when z' increases, that is, when the point, of contact recedes indefinitely from the mirror, this 

 tangent plane approaches to the limiting position 



^=/', (M') 



X 



which is evidently parallel to the consecutive ray (K') ; and the angle (P) which it makes with 

 this limiting position, is given by the formula 



fan p_ (gi — ga)-/^ 



that is, if we put f zz tan. L, 

 or, finally. 



tan p- (ti— g»)-sin. Z. cos. Z. 

 lan. x-- ^ _ (^ ^_ j.j,3 'L + ^^. sin. »Z) ' 



tan.P = y, (NO 



