VOL. XXIII.] PHILOSOPHICAL TRANSACTIONS. IQ 



line BD =/, whence ab = z — /, and bl = ^ r -\-f. Now le^ = ^rr -f 

 rx •\- ff, and pl* = 4. rr -f- 'rx. Therefore le^ — pl'^ = bd"^, which is the 

 latter part of the same theorem. 



The nearer the point f approaches to the point a, or to the vertex, in which 

 the perpendicular cuts the curve, the nearer also the point l approaches to the 

 same. Therefore when p coincides with a, and so the ordinate pd vanishes, 

 then the minimum itself lies in the axis ak, and will be equal to the semi- 

 parameter ; that is, in this case n •=^ \r only ; the absciss x belonging to the 

 vanishing ordinate then also vanishing. If therefore al = n = 4-r, taking the 

 point D between a and l, make ad = a* ; then there arises pl*^ = \rr -\- xx, 

 and therefore fl^ — al'^ = xxy that is, pl* — al'^ = ad^ As it is theor. 2, 

 lib. 7, Conic. De la Hire. 



Secondly, let there be a certain curve of a superior parabolic order, whose 

 equation is 



8p-8 y 2^ 2p-2g iq-p 



rP-9x9 = 2/P ; then yy z= r ^ X -a? ^ ; therefore lyy = - r ^ x ^ x. 



Now if we substitute this value instead of 2yy in the general equation, which 

 deterniines z to be an extreme, we shall have from thence 



2p-2q 2q-p 2p-2q 2q-p 



n^:^-r ^ X ^ -\- ^i therefore the subnormal is dl = -r ^ x ^ , 

 p P 



Now this is easily applied to any of these curves, if the indices p and q are 

 rightly expounded according to the nature and genius of each curve. 



Thirdly, let it be supposed that the curve is an ellipsis, of which ak is half 

 the greater axis. Now it follows, from its equation, that 2yi/ =z rx — 



; whence rx — — - — 2nx + ^^^ = O, and n = - -{• x ; 



therefore — — = dl the subnormal. Now if, instead of the ellipsis, a circle 



were substituted, by proceeding with the equation in the same manner, we 

 should find dl = r — x, making r the radius of the circle. 



But let us return back to the ellipsis, another of whose properties may be 



derived from hence, as was done in the parabola. Make bd = f, whence 



AB = » - /. Then we shall have le'^ = (lb^ + eb^ = ) !I - !If 4. !Ii* 

 + #+ rx - ^f -f; and pl^ = (pd^ + ld^ =) rx - "f V^ - 



VI jf."!^. Therefore le^ - lp' = /)^~ C Now this is theor. 6, lib. 7, 



Conic. De la Hire. 



For that great geometrician requires, that it may be ^ : r :: -^^ — a? : ld, the 



d2 



