VOL. VIII.] PHILOSOPHICAL TRANSACTIONS. 107 



this great surge there are some few lesser surges seen, which are gradually less 

 towards both the east and west. 



FurtJier Solutions of AUiazerCs Problem \ being a Continuation of the Extracts of 

 Letters from M. Huygens and M, Slusius. N° 98, p. 6 140. Translated 

 from the Latin. 



j4nd \st, M. Huygens, in a Letter of April 9, 1672, says: The following is 

 the compendium which I discovered for the first construction, communicated to 

 you before. Drawing AT parallel to CB (fig. 1. pi. 4,) which being bisected in 

 V, gives the point through which one of the opposite hyperbolas ought to pass, 

 the asymptotes of which were found to be YM, MN. — But the construction 

 which obtains in all cases is as follows: Let ED (fig. 2,) be the given circle, 

 whose centre is A; also B and C the given points. Drawing the lines AB, 

 AC, make the following lines proportionals, viz. BA, the radius of the 

 circle, and FA ; also CA, the radius of the circle, and GA. Then join FG, 

 and bisect it in H; through which draw LHK, MHN intersecting at right 

 angles, the former LHK being parallel to that which bisects the angle BAG. 

 Then are these the two asymptotes of the hyperbolas to be described through 

 the points F and G, and one of which will pass through the centre A ; and 

 their intersections with the periphery of the given circle will be the points of 

 reflexion sought. 



Again by Slusius, who, in consequence of the above, writes as follows. — The ex- 

 cellent Huygens's construction is very simple and ingenious. And he has well 

 observed how the equilateral hyperbola may be extended to all the cases, which 

 as I before hinted, immediately offered itself in the case of a right angle. Also 

 that one ellipsis of an easy construction, might be selected out of the infinite 

 number which might be used : but it is tedious to dwell so long on the same 

 problem. But one thing still remains of no unpleasant speculation : that is, 

 since the sections which, with the given circle, are employed for the solution of 

 the given problem, cut it in four points, of which only two can serve for the 

 reflexion : it may be inquired, what problem is solved by the two others; and 

 how the proposition may be expressed that shall include all the four cases; and 

 whether these four cases occur when the given points are equally distant from 

 the centre. 



M. Huygens employs no other than my analysis, which admits of a para- 

 bola only in one case ; as appears from the two equations for the hyperbola about 

 its asymptotes, which I before sent you ; 



viz. 2 zbaa — 2 znae — qqba -\- qqne = bzqq — "Z-qqe, 

 and bzqq — 2znae — qqba -|- qqne =z2zbee — zqqe: • 



P 2 



e> 



