ler 





0,9' 



(C(»)) 



values whicli satisfy the relation (li"") at the beginning of the present paragraph ; and which 

 shew, by the principles there laid down, that the density at the caustic surface is proportional to 

 the following expression : 



that is, in passing from one point to another upon such a surface, or from' a point upon one 

 caustic surface to a point upon the other, the density of the reflected light varies directly as the 

 density at the mirror multiplied by the product of the two focal distances, and inversely as the 

 difference of these distances multiplied by the square root of the radius of curvature of the 

 caustic curve. We see also that the definite integral (Q""), which represents the area of the 

 heartshaped curve that we considered in [68], is equal to the first term of the development 

 (BW) for StO, 



n =48trii*-». tan. i;'./oV(l— «')• V2=4'S'o,o./o,o.r^; (Et*)) 



on which account we may call that curve &pijcnoid, because if r be given, its area is propor- 

 tional to the density at the caustic surface divided by the density at the mirror. 



[70.] The expression that we have just found for tlie density at a caustic surface, becomes 

 infinite in two cases, which require to be considered separately ; namely, first when i z=. 0, that 

 is, at the intersection of the two caustic surfaces, which, as I have shewn, reduces itself to a 

 finite number of isolated points, the principal foci of the system; and secondly, when C = 0, 

 that is, when the radius of curvature of the caustic curve vanishes. A point at which this latter 

 circumstance takes place, is in general a cusp upon the caustic curve ; and the locus of these 

 points forms in general a curve consisting of two branches, each of which is a sharp edge on 

 one of the two caustic surfaces. These cusps are also connected by remarkable relations, with 

 th.e pencils to which the caustic curves belong ; on which account we shall reserve the investiga- 

 tion of the density at such a cusp, until we come to treat more fully of the developable pencils of 

 the systenj. 



[71.] Let us then consider the points wlwre the interval (i) vanishes, that is, let us investigate 

 an expression for the density at a principal focus. In this case we have by the XH'". section, 

 the following approximate formula; : 



X = A»^ -\r 2i5a/3 + C/3% a = — {« 7 .p.^j, 



yz=B<c' + 2C«/3 + D/i\ i = — {/3 3 



(.r, (/) being the coordinates of the point in which the near ray intersects the plane of aberration ; 



VOL. XV. c c 



