Royal Society. 503 



terms of elliptic functions of the first and second orders, and there- 

 fore of having its numerical value calculated from the tables of Le- 

 gendre. The theorem resulting from this reduction, when applied 

 to the particular case of the oscillation of the cylinder, gives an ex- 

 presMon for the time of oscillation, through any arc, of a pendulum 

 having a cylindrical axis. If the diameter of this axis be assumed 

 infinitely small, the case becomes that of a pendulum osciUatnig on 

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

 formula 



where f(c-) 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 arcs, 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 v/A\jump or slip 

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

 the vertical pressure assumes a negative value ; and wliether 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 slipi\ 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 coefficient of friction by the 

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



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

 take place in any revolution is determined by ascertaining whether 

 the maximum value of y 'i that revolution be or be not greater 

 than the coefficient of friction. All tiiese circun.stances are investi- 

 gated on the supposition that the centre of gravity ol 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 knite-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- 

 nir-er of the axis be assumed =0, the case will be arrived at ol the 

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

 symmetrical body revolving about a fixed horizontal axis, Iriction 

 being neglected. 



