or DEFLECTIVE FORCES. 151 



Let AB and CD be two parallel verti- 

 cal ordinates at a constant evanescent 

 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, 6; 6— a, c; AE, u; and EC, v: then BFzz 

 ^/(u^ + c^\ and since CD-EF=EF-AB, FDzz VC^^-f 

 c^). But the velocities at B and F are as s/a and ^/&, 

 and the elements BF, FD being supposed to be described 

 with their velocities, the time of describing BD is V 



f J + s/l — - — J ; which must be a mmimum, and its 



2udu 2i;dt; 



fluxion must vanish : or^ ,, i TT+.T^TTi H 



2v'(a|wM + ccp 2^(h^vv + cci) 



=0; but since AC, or u + v, is constant, du + dvzzO, or 

 duzz — dv ; therefore 



^/as/(uu + cc) s/ b s/ (vv + cc)' 

 Let the variable absciss GA be now called ^, the ordinate 

 AB, y, and the arc GB,z, then u and v are increments of 

 X, and BF and FD of z, when y becomes equal to a and h 



respectively ; we have, therefore, -, the same in both 



cases, so that it may be called -, and-, or -j-= — ^. Now 



a % dz a 



in the cycloid the chord of the generating circle must be 



always a mean proportional between the verse sine y and 



the radius, since, in article 287, CDzz v'(AD.BD) and the 



arc z being perpendicular to that chord, its fluxion, by 



similar triangles, is to that of the absciss a?, as the diameter 



to s/y : therefore the cycloid answers the condition in 



every part, and consequently in the whole curve. 



