108 PHILOSOPHICAL TRANS ACTIONiS. [aNNO 1673. 



And I added, that by a small variation, viz. substituting for example, for qq its 

 value aa -^ ee, there may be found an infinite number of hyperbolas and ellipses^ 

 which, with the given circle, would solve the problem. Now, in the former of 

 these equations, for bzqq put its value, and it gives zhaa — 2znae — qqba -\- 



7 qqa qqe , 2nae qqne 



qqne = bzee — zqqe. or aa --^— = ee — —- i^ — . 



■'■' .11 2 b * b zb 



And this is the very equation, which that learned gentleman has constructed 

 with so much ease and ingenuity. 



^gain, writes M. Slmins, I happened lately on the following construction, 

 which I submit to your judgment and censure, as I think a shorter can hardly be 

 found. Let the given points be E, B, (fig. 3,) and the circle whose centre is A; 

 join EA, B A., cutting the circle in Fand C ; make the three proportionals EA, 

 F A, VA, and the three others BA, CA, XA: then VX being joined, and pro- 

 duced at pleasure, with the vertex X, diameter VX, and latus rectum equal to it, 

 describe the hyperbola XP, whose ordinates to the diameter VXG are parallel to 

 AB. For this satisfies the proposition in the case of the convex speculum, and 

 its opposite in the case of the concave one. As to the asymptotes, they will be 

 easily found by producing VX till it meets EB produced in L; then bisecting 

 VXin I, and taking LD equal to LI; for joining DI, it will be one of the 

 asymptotes, which the other meets perpendicularly at L 



But perhaps it will not be unacceptable to you to know how I arrived at this 

 construction. This then I deduced from my former analysis as follows : the 

 same things being given as before, demit AO perpendicular toEB, (fig. 4,) and 

 let P be the required point, from which draw PR perpendicular to AO. Put 

 AO = h, EO = z, OB = d, AP = q, PR = e, AR = a ; then is easily deri- 



, ,1 . ,. 2 zdae + 2 bbae — 2 bqqe , qqa ^ • % ^ 



ved this equation, "hZTfl 1- ee = aa — ^-, which may be 



, 1 • , i^i zdae + bbae — bqqe qqa 



changed into these ^~bd — ~ ^^ ~ '^99 ~~ 2 P 



1 zdae -{-bbae — bqqe ^^ qqa 



^"^ Tb^Tbd h ee — ^qq -j. 



I formerly sent you the construction of this last; and M. Huygens has sent 

 you the construction of the other. As to this one indeed, though I at once 

 perceived it, yet I rather neglected it, as the construction seemed to be difficult. 

 But I find I had been deterred by a groundless fear, as I have lately found it 

 lead me to this construction, which I now send you. To abridge the calcula- 

 tion, make z — d z= k, and zd~\- bb = bm; then it will be ee -f- r- — ^ 



qqa , , ,. q"^ + mmaa — 2 qqma 1, lU -j -i. • 



=: aa — ^-; and adding -^—^^ rr — on both sides, it is ee -f 



2mae- 2jqe_ _^ q^ + .nmaa- 2 gqma ^ ^^^^ .^^ ^^^ ^^^^^^^ ^^ ^ _^ ma-jq^ ^^^^^^ 

 qqa , q^ + mmaa ~ 2 qqma ,. , . ... ,. ,, jj , 



to aa — -^ h 7-7 ^ — ; which gives this proportion, A A : kk -f mm:: 



