300 PROFESSOR TAIT ON THE ROTATION OF A 



so that the substitution in (46) of the value of e from (47) gives 



BYaU-AQa = Yay (49), 



an extremely simple equation to determine a. It is curious to remark that this 



is the equation of motion of a simple pendulum, disturbed by a force constantly 



perpendicular to the cone described by the string, and proportional to the rate at 



which the area of the surface of the cone is swept out by the suspending cord. 



When A = it becomes that of the undisturbed motion,* and gives a number of 



curious theorems relating to the curvature of the general path of a simple 



pendulum. These we need not at present consider; though we may mention 



that the corresponding equation for the motion of Foucault's pendulum may be 



written in the form 



Y«(« + /3) = eVajS, 



where /3 is a vector known in terms of t. 



55. If we suppose a determined in terms of t from this equation, (46) gives s 

 in the form 



Be = (A - B)Qa. - V . yfa.it . 



This equation may be obtained, even more simply, from (47). 



56. But, without finding either a or s, we may deduce various facts connected 

 with the motion. Thus operating on (46) by S . e, we get 



BSeI = S . say = Sya , 



which gives 



Be 2 = 2S 7 a + C (50). 



But, by operating on the same equation by S . y and integrating, we have 



BS 7£ -(A-B)ftS 7 « = C 1 .... (51), 

 which may be written in the form 



Sep/ = Sytpe = C x (51)'. 



By (50) and (51) 



B.' = ^^ + C, 



so that g is a vector of a fixed sphere, of which however the centre is not at the 

 fixed point. 



* If m be the mass of the pendulum bob, a the vector representing the string, 2T its tension, and 

 y the acceleration due to gravity 



mot, = my — 21 Ua , 

 or, eliminating ®, 



Vaa = Vccy. 



It is well to observe that this is the equation of motion of a pendulum bob, acted on by no forces, 

 if — y be the acceleration of the point of suspension. 



