144 



that is, at the positive end of the diameter ; and which crosses the curve in two other points, ' 

 equally distant from the diameter, and having for coordinates, 



a — i.r. P-^, tan. v', b = :±z \/{i.r. sin. 2i/). 



And the whole area of this heartlike curve is equal to the following definite integral, 



U = iP-K tan. v'.fx/{i'.r' — b*).db, (Q"") 



the integral being taken from 6 = 0, to b = Vi.r. In the next paragraph we shall return to 

 this definite integral, and shew its optical value. 



[69.] But the preceding calculations only shew how the density varies near the caustic sur- 

 face ; to find the law of the variation at that surface, we must reason in a different manner. 

 For if the infinitely small rectangle on the plane of aberration, which we have considered in the 

 preceding paragraph, have one of its corners on the caustic surface, we can no longer consider the 

 density as uniform, even in the infinitely small extent of that rectangle. But if we consider the rays 

 that pass within a given infinitely small distance {dr) from a given point upon the caustic surface, 

 for example, from the focus of the given ray, we can find the space over which these rays 

 are diffused upon the perpendicular plane at the mirror ; and this space, multiplied by the 

 density at the mirror, may be taken for the measure of the density at the given focus, not 

 as compared with the density at the mirrror, but with the density at other points upon the 

 caustic surface. 



To calculate this measure, let us consider the following more general question, to find the 

 whole number (0!-''^) of the near reflected rays which pass within any small but finite distance 

 (r) from the focus of the given ray, and the space {S'^'''>) over which these rays are diffused, on 

 the perpendicular plane at the mirror. This question evidently comes to supposing the little 

 circular sector (r^. i|/) of the preceding paragraph completed into an entire circle, and conse- 

 quently may be solved by integrating the formulae (E"") (F"") of that paragraph, within the 

 double limits afforded by the equation of the circle on the one hand, and by that of the section of 

 the caustic surface on the other ; since it is easy to see that only a part of the little circular area 

 (■a.r' ) is illumined, namely, the part which lies at that side of the caustic surface, towards which 

 is turned the convexity of the caustic curve. 



But as the formulae of the preceding paragraph were founded on the developments (T'") 

 which, as we have before remarked, become illusory when the polar radius (r) approaches to 

 become a tangent to the caustic surface, (a position of that radius which we are not now at 

 liberty to neglect, ) it becomes necessary to investigate other developments, and to transform 

 the double integrals (E"") (F"") of [68] into others better suited for the question that we are 

 now upon. And to effect this the more clearly, it seems convenient to consider separately the 

 four following problems: 1st, to find general expressions for the coefficients uW, ujW, which 

 enter into the developments (T'"), and to examine what negative powers they contain of the 

 sine of the polar angle (v); 2d, to eliminate these negative powers, and so transform the two 

 series ^T'") into others which shall contain none but ascending powers of any variable quantity; 



