VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 35^ 



pursued; we shall therefore at present have recourse to this way of considering 

 curves. And first we shall show how easy it is in this method to derive 

 the more complex curv^es from the simple ones. 



Sect. 1. Let L and /, fig. 1, pi. Q, be two points in the curve b/l very near 

 each other; and lo an arc described with the centre s, and perpendicular to sl; 

 then l/ will be as the moment of the curve, and lo as the moment of the 

 radius sl: and if there be given the ratio of l/ to lo, or to lo in the distance 

 SL, there will be given the equation of the curve to the centre s. Let lp, Ip 

 be tangents to the curve at the points l, /; on which draw the perpendiculars 

 sp, sp. In like manner, on all the tangents of the curve let perpendiculars be 

 drawn from the given point s; and there will be constructed a curve passing 

 through all the intersections of the tangents and perpendiculars. Of this curve 

 the elementary triangle ¥np will be similar to the triangle lo/, which will there- 

 fore be given from the given curve b/l. For because of the equal angles snp, 

 PWL, and the right angles spn, spl, the triangles spn, pul, will be equiangular, 

 and therefore pw : pn :: L.n : sn :: lo : lo; and because of the equal angles pwp, 

 snL, Lol, the triangles vnp, swl, lo/ will be similar. Since therefore the ratio 

 of l/ to lo is the same as that of pp to pn, and of sl to sp, it is manifest 

 that, having given sl and the ratio of l/ to lo, there will be given the ratio of 

 pw to pn, and the line sp, and therefore the curve dp/?. In the same manner, 

 from DP there may be constructed a third curve, and from that a fourth; and 

 so, proceeding in this manner, an infinite series of curves will be produced, 

 which will all become known from the given one. Now if ln and In be drawn 

 perpendicular to the radii sl, s/, meeting in n; and through all such points of 

 concourse of the perpendiculars the curve en be described ; that will be a curve 

 from which bl may be deduced, after the same manner as dp and bl were con- 

 structed. In like manner, from en may be constructed another curve, and 

 thence on this side likewise an infinite series of curves may be constructed. 



2. But of all the curves thus produced, the simplest will be those in which 

 l/ is to lo in the ratio of some power of the radius : so that if a be a given 

 quantity, r the radius of the curve, and n any number, then if l/ : lo :: a" : r% 

 it will give their general equation. And all these v/ill have an apsis when 

 r = a, because then l/ = lo. To investigate the equation of the curve dp; 

 since in bl it is, l/ : /o :: a" : r" :: r : sp = 



» I n n 



:: a""^^ sp""*"^ : sp :: a""^' : sp'"*"' .•: pp : pn; therefore if s represent the 



moment of the curve, {/ the circular arc described by the radius from the 

 centre s, and r the corresponding radius, whatever the curve be whose equa- 



