360 PHILOSOPHICAL TRANSACTIONS. [aNNOJ7I8. 



The equation of the circle, 1. 

 of the epicycloid, 2. 

 of the second, 3. 

 of the third, 4. 



of any one, 5. ^ - y :*• an : rn. &c. 



Here it may be observed in general, that all those which have the denomina- 

 tors of their indices even numbers, are capable of perfect rectifications; and 

 since any one is to that before it, as 1 to 1 — w, it will appear on con- 

 sideration that the length of any curve will be 



1 . 1— 2m 1 — 4»i l — 6m „ ,• • i • .« • 



= ; X r-^- X ■; — — X -r-=- &c. X SB, contmumg the series till the 



1 — tn 1— 3m 1 — 5m 1— 7m ' * 



fraction be reduced to nothing. But if the denominator of the fraction be an 

 odd number, the curves will be incapable of perfect rectifications, and any one 

 of these arcs will be incommensurable to any other, and to the wholes, and to 

 right lines, and to circular arcs : yet they may all be expressed by circular arcs 

 and right lines: and the total length of any curve will be to the semicircle, 



as ; X :; — r— X - — t" X &c. to unitv. Lastly, if the areola described by 



1 — m 1 — 3m 1 — 5m •' •" . •' 



a body revolving in any one of these be taken as constant, that is, if ri/ = 1, 

 the subtense of the angle of contact, to which (because of the time being given 

 when the area is given) the centripetal force tending to s is always proportional, 

 will be reciprocally as the power of the distance whose index is 2m -f- 3. And 

 this is no contemptible property of these curves, that in all of them the centri- 

 petal force tending to s, is reciprocally as some power of the distance : which 

 is the most simple and useful law of centripetal forces, in searching into 

 nature. 



5. Of all the curves in which s : y :: a" : r% the right line is next to be 

 considered (which is indeed improperly called a curve) the point s being without 

 that right line, fig. 4. In this line, because of the similar triangles pj&n, pbs, 

 if BS = a and sp = r, it will be * : ^r :: r : a. By the direct method nothing 

 can be constructed from the right line, except the point b: but by the inverse 

 method, from the concourse of the perpendiculars pl, p/, a curve may be con- 

 structed, whose index will be equal to , if m be the index of the curve bp. 



^ 1— m 



m 



For if the index of bl be n, then it will be m = , and therefore n = , 



n+l' l—m 



Hence in this case, since m = — 1, it will be n = — 4-, and the equation of 

 the curve bl will be * : ^ :: A : at, which is an equation of the parabola re- 

 ferred to the focus. From this construct another, by making the angle lsn 

 = lsb, and erecting ln perpendicular to sl meeting sn in n. Now be- 



