being taken into account in forming the equations of motion, and its 

 intensity supposed such as to counteract that part of the total angular 

 velocity of the axis which is perpendicular to the given plane. The 

 equation which determines the motion of the axis is shown to be 

 identical with that of a circular pendulum, and the motion con- 

 sequently one of oscillation, the mean position of the axis being that 

 in which it approaches, as close as the conditions of the question 

 permit, to the line drawn through its centre of gravity parallel to 

 the earth's axis, and in which it rotates in a direction similar to that 

 of the earth's rotation. Similar reasoning establishes the second 

 part of the theorem, which is theoretically true whether the gyro- 

 scope be set rotating or not. This result is, however, in practice 

 modified by the effects of friction ; but when a rapid rotatory motion 

 has been impressed on the gyroscope, it acquires a stability which 

 enables it to overcome to a great extent these effects. 



December 13, 1860. 



Major-General SABINE, R.A., Treasurer and Vice-President, 

 in the Chair. 



The Right Hon. the Earl of Sheffield, and the Right Hon. Spencer 

 Horatio Walpole were admitted into the Society. 



The following communications were read : 



I. " On an Extension of Arbogast's Method of Derivations." 

 By ARTHUR CAYLEY, Esq., F.R.S. Received October 18, 

 1860. 



(Abstract.) 



Arbogast's Method of Derivations was devised by him with a view 

 to the development of a function <f>(a + bx + cx 2 + ...), but it is at 

 least as useful for the formation of only the literal parts of the co- 

 efficients, or, what is the same thing, the combinations of a given 

 degree and weight in the letters(a, b, c, d, . . .), the weights of the 

 successive letters being 0, 1, 2, 3, &c. Thus instead of applying the 

 method to finding the coefficients 



6a 2 t> 2 , &c., 



