62 J. G. Barnai'd on the Gyroscope^ 



For sin <9 -r; is the liorizontal, and -r- the vertical, component 



at at 



of this velocity. Calling the first t?^, and the second Vx>^ and 

 the resultant v^^ and calling cos 5— cos «, (which is the true 

 height of fall) A, those equations may be written 



Cn h , . 



This velocity v^ (as a fnnction of the height of fall) is exactly 



that of the compound pendulum^ and is entirely independent of the 

 axial rotation n. Hence, (as we might reasonably suppose) ro- 

 tary motion has no power to impair the work of gravity through 

 a given height^ in generating velocity ; "but it does have power to 

 change the direction of that velocity. Its effect is precisely that of 

 a material undulatory curve, which, deflecting the body's path 

 from vertical descent, finally directs it upward, and causes its 

 velocity to be destroyed by the same forces which generated it 

 And it may be remarked, that, were the cycloid we have de- 

 scribed such a material curve^ on which the axis of the gyroscope 

 rested, without friction and without rotation^ it would travel along 

 this curve by the effect of gravity alone, (the velocity of descent 

 on the downward branch carrying it up the ascending one,) with 

 exactly the same velocity that the rotating disk does, through the 

 combined effects of gravity and rotation. 



Equation (a) expresses the horizontal velocity produced by 

 the rotation. 



If we substitute its value in the second, we may deduce 



V^ OT —= \~h 



dl O' A'^ sin^^ 



If we take this value at the commencement of descent, and 



before any horizontal velocity is acquired^ (making h indefinitely 



small), the second term under the radical may be neglected, and 

 the first increment of descending velocity becomes 



cisely what is due to gravity, and what it tvould he were there no 

 rotation. 



Hence the popular idea that a rotating body offers any direct 

 resistance to a change of its plane, is unfouujicd. It requires as 

 little exertion of force (in the direction of motion) to move it . 



itself; -while its power of generating living force by working through a given 

 height, cannot be impaired. 



Had we considered ourselves at liberty to assume them, however, the equations 

 might have been got without the tedious analysis by which we have reached them- 



