*. 162 



function vanishes. It is evident that at these points (sin. v) is small ; if then we change 

 r. sin. (v — i/) to — r. sin, v', the condition € = becomes by (YW) 



= sm. « + (5^ + (^ ) . (^- P-^- tan. «?. r. 



that is, in the notation of paragraph [61. 3 1 



2i»C. sin. v=:(AC — B')r {Zm ) 



which, by the same paragraph, is the equation of the osculating circle to the section of the 

 caustic surface ; from which it follows, that in this approximation, the function (S) is, for any 

 other point upon the plane of aberration, the distance of that point from the osculating circle 

 just mentioned, measured in a direction parallel to the normal of the caustic surface. More 

 accurately, if we put 



r. sin. e=y. sin. t/, r. sin. (« — u') = x". sin. v', • (AW) 



x", ^", will be the oblique coordinates of the point r, v, referred to two axes in the plane of 

 aberration, of which one touches the caustic surface at the focus of the given ray, while the 

 other is inclined to this tangent at an angle = u' ; and the equation (Y'^)) will become 



'-{^Pc-)--"' """) 



;=y. ■ "^'^-« 



which shews that (£) vanishes for the points of a parabola, which has its diameter parallel to the 

 axis of y, and has contact of the second order with the section of the caustic surface; and that 

 for any other point upon the plane of aberration, (S) is equal to the distance from this parabola 

 measured in a direction parallel to its own diameter, and then projected upon the normal. If, 

 therefore, in the developments (Wt^^), we change r. sin. v, r sin. {v — 1/), totheir values y". sin. v', 

 x". sin. v' (A(") ; if we then eliminate y". sin. v', by changing it, in virtue of (BP'), to the binomial 



AC-B\ 





and develope every fractional power of this binomial according to the ascending powers of x", and 

 the descending powers of £, we see that the new developments will contain no negative powers of 

 this latter variable, except those which arise from the terms that we rejected in effecting the 

 summations (X^^') : and I am going to shew, that if in place of the parabola S = 0, which has 

 contact of the second order with the section of the caustic surface, we take that section itself, 

 whose equation referred to the coordinates (x", y") is of the form £ '=0, in which 



€'=s-2^^-. [o]-^''+») (£^)- x'"'+' = (y'_yg sin..', (ca>) 



^"0 being the ordinate of the section, and %' the distance from that curve, measured in a di- 



