VOL. XXIlI.j PHILOSOPHICAL TRANSACTIONS. 21 



it appears that this is the exemplar, since c : r — c ::/:--■— /, and there- 

 fore, according to the definition, ^ — / X / is equal to the exemplar. Now 



this is theor. 7, lib. 7, of De la Hire's Conies. 



Again, when the point p coincides with a, because of x vanishing with the 

 ordinate then vanishing, there is left n = ^R, and al is the greatest of all the 

 lines that can be drawn from the point l to the ellipsis, and al^ — pl* = 



— — XX :=. X.O the exemplar applied to ad or x. And the same as theor. 



4 of said lib. Conies. 



But it ought to be observed at the foregoing case, (which should have been 



mentioned before) when we found w = - •\- x , that n <; -i^r. For 



en •\- V.X •=. -i-Rc -j- ex ; and because h >« c, therefore rjc >► cx^ and there will be 

 left en -< 4-Rc, or n >- -I-r. 



Now as the matter is performed in the ellipsis, so in the same manner it might 

 be performed in the hyperbola, and the least lines may also be determined in 

 this curve. But there is such a connection between these two curves, and the 

 transition from one to the other is so easy, that the labour may seem unnecessary 

 even to novices. Therefore nothing more remains to determine the subnor- 

 mal, than that the sign — may be changed into +. For since in the hyper- 

 bola it is lyy = rx -j -, and n = - -f- .r -f — ; (the general equation) 



there remains dl = — I . 



2 ^ y 



Let it be conceived fourthly, that the curve msn (drawn on the other side 

 t,he figure) is one of the hyperboloids, whose asymptotes are ak, kh, and the 

 right line sr an ordinate to the asymptote kh ; make sr = ^, sp =. z, kr = 

 ar, KP = w, which here must needs be less than x^ as will appear on considera- 

 tion. The equation proper to the curve is yi^x^ = ns'P^ instead of which, be- 

 cause of r and s being determinate quantities, may be written y^ == jr-?, and 



therefore y'^ •=■ x ^ , and lyy = — —xx ^ . Hence, since it is zz ■= yy 

 ■\- XX — 2nx -}- nn, for an extreme we have 2yy -|- 2xx ~ 2nx = O ; that is, 



-XX ^ -f- 2xx = 2w.r, and n = x '- -x ^ ; therefore (a: — n =) 



-ar P is the subnormal pr. 



Lastly, let us conceive the curve afg to be a primary cycloid, and let the 



