34 



OF MOTION CONFINED TO GIVEN SURFACES. 



the curvature is continued, no velocity is 

 lost in the change of direction ; for let AB 

 be the thread or its evolved portion, the 

 body B, if no longer actuated by gravity, 

 C would proceed in the circular arc with uni- 

 form motion (240), consequently no velocity is destroyed 

 by the resistance of the thread, nor by that of the surface 

 BC, which can only act in the same direction, perpendicular 

 to the direction of the moving body. 



259. Theorem. If a body be suspended 

 by a thread between two cycloidal cheeks, it 

 will describe an equal cycloid by the evolu- 

 tion of the thread (208) ; and the time of 

 descent will be equal, in whatever part of 

 the curve the motion may begin, and will be 

 to the time of falUng through one half of the 

 length of the thread, as half the circum- 

 ference of a circle is to its diameter. And 

 the space described in the cycloid will be al- 

 ways equal to the versed sine of an arc which 

 increases uniformly. 



For since the accelerating force, in 

 [he direction of the curve, is always 

 (O the force of gravity as AB to BC, 

 -5- or as BC to the constant quantity BD, 

 it varies as BC, or as its double, CE, 

 *hc arc to be described (208). If 

 therefore any two arcs be supposed to 

 be equally divided into an equal num- 

 ber of evanescent spaces, the force will be every where as 

 the space to be described ; and it may be considered for 

 each space, as equable, and the increments of the times, 

 and consequently the whole times, will be equal. Suppo- 

 sing the generating circle to move uniformly, the velocity 

 of the describing point C will always be as CD (209), or, 

 since AD : CD : : CD : BD, and CD=v' (AD.BD), as 

 ^AD ; but the velocity of a body falling in DA, or de- 

 scending in FC, varies in the same ratio (232, 230, 258) ; 

 therefore if the velocity at E be equal to that which a body 

 acquires by falling through GE, the describing point C will 

 always coincide with the place of a heavy body descending 

 in FCE ; and the velocity of the point of contact D is half 

 that of Cat E (209), it would therefore describe a space 

 equal to GE in the time of the fall through GE (232), and 

 ■will describe FG in a time which is to that time as FG to 

 GE, or as half the circumference of a circle to its diameter. 



and this will be the time of descent in the cyclwdal arc. 

 And since FCziaDB — 2BC, FC is equal to the versed sine 

 of the angle CBD, to the radius 2DB ; but /.CAD increas- 

 ing uniformly, its half, CBD, increases uniformly. And if 

 the motion begin at any other point, the velocity will be in 

 a constant ratio to the velocity in similar points of the 

 whole cycloid. It is also obvious that the arc of ascent 

 will be equal to the arc of descent, and described in an 

 equal time, supposing the motion without friction. 



260. Theorem. The times of vibration 

 of different c^'cloidal pendulums are as the 

 square roots of their lengths. 



For the times of falling through half their lengths are 

 in the ratio of the square roots of these halves, or of the 

 wholes. 



261. Theorem. The cycloid is the curve 

 of swiftest descent between any two points 

 not in the same vertical line. 



GAE r 



VD 



Let AB and CD be two parallel ver- 

 tical ordinates at a constant eva- 

 nescent distance, in any part of the 

 curve of swiftest descent, and let a 

 third, EF, be interposed, which is 

 always in length an arithmetical mean between them, and 

 which, as it approaches more or less to AB, will vary the 

 curvature of the element BFD. Call AB, a ; EF, b ; b — a, 

 c; AE, u; and EC,«; then BFrZv' ("«+«), and since 

 CD— EF=EF-AB, FD=v^(dii+cc). But the velocities 

 at B and F are as i/a and i/b, and the elements BF, FD, 

 being supposed to be described with these velocities, the 



/ Mu+cc \ /iT-J-rcA 



time of describing BD is v' 1 / "*" ^ ( — 7, — J ' 



which must be a minimum ; therefore its fluxion vanishes, 



2i(u . aKi , . . • 



or ; : — -A rr, : — TT— " i °"t ^mce AC 



2v'(a(i'«+"j) 2^/(0 (««+«)) 



or u+v is constant, M-f-i~o, or «:^ — b ; therefore, 



-— :z — rr. Let the variable abscis* 



^{a{tm+cc)) ^{b{vv+cc)) 



GA be now called x ; the ordinate AB, y ; and the arc GB 

 z ; then u and v are increments of x, and BF and FD of z, 



when V becomes zza and b respectively ; and -.is the 



same in both cases, and is^ therefore constant, or = — , and 



.-^y 



Now in the cycloid v'^ 's always xixn 



