J04 PttlLOSOl'HiCAL TRANSACTIONS. [aNNO 1073. 



or b '.y — n:: e:y — a. Therefore —, = w = —^ — . Hence 2zbaa-^ 



J J t) — e «/ za — qq 



2 znae — qqha -f qqne = zhqq — "^qq^, which is the very equation to Alha* 



zen's problem above formed : or, in the second case, -7— = 7/ = — ^^ — , 



^ ' b + e -^ 2za — qq^ 



or Izhaa + Iznae — qqha — qqne = zhqq + zqqe. 



And these are the problems commonly proposed concerning the point of 

 reflexion in which the distance of the given point D is supposed to be finite. 

 But the analysis will be easier if it be supposed infinite. For bisecting CA in 

 G, it appears from the property of the three harmonical proportionals, DA, 

 CA, BA, that the three DG, CG, BG will be geometrical proportionals, sup- 

 posing the point D at any distance whatever. Therefore if it be supposed 

 infinite, BG will become nothing, and the point B coincide with G. Conse- 

 quently AB will be always equal to BC : therefore CA = 2?/, and the rectangle 

 CAI equal to AE^ will give in symbols 2ay = qq, or 3/ = —^i and since the 

 distance of the point D is supposed infinite, E D will be parallel to AC. There- 

 fore, if there be required the reflected ray parallel to AL, because in that case 

 a and ?/ coincide, it will be a = ?/ = ~, or aaz=.±.qq. If it be required to be pa- 



1-allel to AK, it will again he, 2ls q-. d r. e \ a — y, and -^^-^^^^-^ = z/= ^, or iqaa 



— 2rfae = ^^ : if it be required to pass through N, it will be as above, ^ -"^ . = 



y sz ^ , and 2 haa + 2nae =: hqq + qqe, which are also equations to the 

 hyperbolas about the asymptotes, unless the point N be supposed to be in AL; 

 for, as in that case n becomes nothing, by taking out of the equation the terms 

 in which n is contained, the remainder will give the equation to the parabola, as 

 we before observed. 



You will not expect, learned sir, that as a I have hitherto given examples for 

 the concave specula, so I should now proceed to the convex. For you know 

 that the analysis is the same in both, and their equations only differ in the varia- 

 tions of the signs + and — . You know that the parabola or ellipse which solv^es 

 the one, will satisfy the other also; and if the hyperbola solve the problem in 

 the convex, the opposite hyperbola will do the same in the concave. Omitting 

 these then, I shall only add, that by the same analysis in concave speculums, we 

 may find their foci, and the spaces occupied by the rays in the axis, for any given 

 distance of the lucid point : but with great facility when the rays are supposed 

 parallel ; which yet I have seen demonstrated by some in a very tedious way. 

 For in the concave speculum EE (fig. 9), whose centre is A, if the extreme 

 ray be supposed as reflected to the axis AR at B, drawing the tangent EC, it 



