of the Acceleration of Gravity for Tokio, Japan. 297 



of the wire to the ball capable of resisting the tendency of the 

 ball to continue its turning motion. If we do this by solder- 

 ing the wire, a smaller kick will result, and will be due to the 

 bending-moment of the wire resisting the turning action. If 

 there were no difficulty of construction, it might be better to 

 get rid of this kick difficulty by making the bob capable of 

 rotating in the plane of swinging about an axis through its 

 centre of gravity. 



The investigation of the general problem of the swinging of 

 a heavy ball soldered to an elastic wire, the upper end of which 

 is attached to a knife-edge, may take somewhat the following 

 form: — Let A be a ball, of mass m and radius a, suspended 

 from a free hinge at B by a chain of n— 1 links, each of 

 length a and hinged to one another, the last hinge being on 

 the surface of the ball. Suppose at any hinge where two ad- 

 jacent links make an angle 6 with one another, equal and op- 

 posite couples act in them of moment c9 tending to bring them 

 into the same straight line. As n is made greater and greater, 

 we approximate more and more nearly to our actual case of 

 an elastic wire. Let n be very great, and fa, fa, . . . $«-i 

 the inclinations of the 1st, 2nd, . . . (n— l)th link to the vertical, 

 the nth. link being the radius of the ball up to the hinge, and 

 its inclination fa. If now we know the mass of the links per 

 unit of length, it is easy to state the values of yjr 1} \jr 2 ,. . . . 

 tysj • • • tyn) the couples acting on each respective link : thus 



care being taken to remember that the form of ty 8 is different 

 from that of ifa or of ty n . 



If the inclination <£ is everywhere very small, we find, if v 8 

 is the velocity of the end of link s, that 



t=<?\(faf + 2fafa cos (fa- fa) 4- (26-3)(<k) 2 } , 



d<f> s 

 where fa means —rr> 



? So that the kinetic energy T of the whole system may at once 

 be written out in terms of the coordinates fa, fa,.** &c., 

 fa, fa,... &c. 

 We can therefore find the partial differential coefficients 



dT dT „ , ^T dT - 



— — , — -, &c, and -T7-, -n-. . . . &c, 

 dfa' dfa ' clfa J dfa> % > 



so as to use Lagrange's equation 



dt\d$) d<f> T 



