36 



knife-edges ; and the time of oscillation is expressed by the simpfr 

 formula ^ 

 <=-F 



TT 



where ^(^^^ represents that complete elliptic function of the first 



order whose modulus c is the sine of half the angle of oscillation. 

 From this formula the times of oscillation through every two degrees 

 of a complete revolution have been calculated in respect to a pen- 

 dulum which beats seconds when oscillating through small arcsy and 

 are given in the form of a table. 



In the second part of the paper, general expressions are arrived 

 at for the vertical and horizontal pressure of the cylinder upon the 

 plane on which it rolls, at any period of a revolution ; and these are 

 applied to determine the conditions under which it will jump, or slip 

 upon the plane. A jump will take place when the expression for 

 the vertical pressure assumes a negative value ; and whether such a 

 jump will or will not take place in any revolution is determined by 

 ascertaining whether the minimum value of the pressure in respect 

 to that revolution be negative or not. The cylinder will slip if its 

 friction on the plane fall short of the horizontal resistance X, de- 

 termined as the necessary condition of its rolling. As the friction 

 is measured by the product of the eoefBcient of friction by the 

 vertical pressure V, it follows, that slipping will take place if 



Y exceed the coefficient of friction ; and whether it will or will not 



take place in any revolution is determined by ascertaining whether 

 X 



the maximum value of y in that revolution be or be not greater 



than the coefficient of friction. All these circumstances are investi- 

 gated on the supposition that the centre of gravity of the cylinder is 

 situated at any given distance from its axis, and that it is projected in 

 any position with a given angular velocity, which angular velocity 

 must be assumed =0, to get the case of an oscillatory cylinder. 

 The investigation determines in this case the circumstances under 

 which a pendulum oscillating by a cylindrical axis, or by knife-edges 

 on horizontal planes, will jump or slip upon its bearings unless other- 

 wise retained. If a finite value be assumed for the angular velocity 

 sufficient to cause complete revolutions to be made, and if the dia- 

 meter of the axis be assumed =0, the case will be arrived at of the 

 pressure upon its bearings of a falsely-balanced wheel, or any un- 

 symmetrical body revolving about a fixed horizontal axis, friction 

 being neglected. 



If the angular velocity of projection be supposed to be that ob- 

 tained by the cylinder when its centre of gravity is at its highest 

 point, the general formula for the vertical pressure assumes a simple 

 ibrm, under which it is readily applicable to the case of the falsely- 

 balanced carriage wheel, a case which assumes a practical importance, 

 from the fact that the driving wheels of locomotive engines are all, 

 by reason of their cranked axles, falsely balanced unless counter- 



