132 TRANSACTIONS OF THE [APR. 11, 



falls, at a rate varying as some direct function of the downward- 

 pull, or weight, and, inversely, as some direct function of the 

 rate of rotation. Oi', calling h the time of horizontality ; w, 

 some direct function of the weight ; v, a direct function of the 

 velocity of rotation ; we have A oo - — /, where / = the effect 

 of friction. The only possible effect of / is to decrease h. 



Y 



Hence, when /" = 0, we still have h co -. Now v and w are 

 independent variables, and direct functions of velocity and 

 down-pull. Hence, h can become infinite only when t; = co , 

 or lu = 0. The first is impossible ; the second is merely saying 

 that a body will stay up infinitely long if no force acts upon it to 

 send it down. 



Second Proof. — (1) I remove the friction at the point of sup- 

 por by causing it to revolve in the same sense, and a trifle faster 

 than the gyrational or precessional movement. (2) I replace the 

 energy lost by friction at ends of the axle, and by the friction 

 of the air. 



The first is easily accomplished by suspending the gyroscope 

 by a twisted cord, whose torsion acts in the same direction as 

 the gyration. 



The second is accomplished by an electric motor driven by a 

 small battery.* For this purpose a form of Bohnenberger is 

 used, the point of support being in a vertical line passing through 

 the centre of gravity of the wheel. A small horse-shoe electro- 

 magnet is attached to the ring of the instrument, and is rapidly 

 driven by four cells of a Smee's battery. The connections are 

 through very large open cups of mercury in which the connect- 

 ing wires revolve. The whole is supported by a fine cord about 

 six feet long, the torsion of which is made alternately to aid und 

 to hinder the horizontal rotation. Starting from rest, the 

 wheel quickly reaches a speed at which the loss and gain of 

 energy are equal ; and, consequently, the rate of motion on the 

 axis is uniform. It will continue uniform so long as the battery 

 is constant ; and, as the experiment occupies but a few seconds, 

 or a few minutes, according to the weight applied, even a bi- 



' Mr. Geo. M. Hopkins was the first to apply electricity to a gyro- 

 scope. 



