VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 36l 



cause w = — -i-, it will be n = — 4-j ^"d the equation of the curve will be 



BN— LN 



s : V '.'. r7 : ai, and bp = — = ^. bn — ln; therefore bn = 2bp + i-n; 



so that this curve is rectifiable. If the series be continued, the equations will 

 arise as before in this order: 



Equation of the right line, s '. y '.: r : a . 



of the parabola, s : i/ :: r'2 : ai. 



of the second, * : i/ :: r's : a?. 



of the third, s : i/ :: i"^ : a^. 



r ■ • ^ i. 



oi any one, s : y •.'. r'^ '.an* 



In this series, the first are the right line and the parabola; whence it appears 

 that half of this series, as well as the former, are commensurable to right lines: 

 and the other half may be exhibited by right lines and parabolic arcs. In all 

 these, the centripetal force at s, is reciprocally as that power of the distance, 

 whose index is 3 — 2m; and therefore is always between the duplicate and tri- 

 plicate ratio of the distance reciprocally. 



6. The equation of the equilateral hyperbola at the centre, is * : y :: r^ : a^; 

 from which, by the direct method, may be deduced series of this kind: 



1. s : i/ '.'. r^ : a^ 



2. s : y M a^ : r^ 



3. s : y :: a'^ : vs 

 A. s : y :: al : r? 



5. s : y :: a*""' : r*" 



Of these curves, those which have the denominators of their indices in this 

 progression, —1, 3, 7, 11, &c. may be exhibited by right lines and hyperbolic 

 arcs; and the rest by right lines and arcs of the curve, whose equation to the 

 axis SB is fxx + yyj'^ = a"^ x'^ — a^ y'^ (the absciss being jc, and the ordinate y) ; 

 and which is constructed (fig. 3) by bisecting the angle bsl, and taking sn a 

 mean proportional between sb and sl. 



The curves which may be constructed from the hyperbola by the inverse 

 method proceed as in this series : 



Of the hyperbola 1. s : y :: i'^ : a^ . 



n ■ • 2 2 



2. s : y :: r"^ : a-j. 



3. s : y :: r"^ : a\. &c. 



Where the curves which have the denominators of their indices in the progres- 

 sion 1, 5, 9, 13, &c. may be expressed in right lines and hyperbolic arcs; and 

 the rest in right lines and the arcs of the curve above explained. 



VOL. VI. 3 A ^ 



