728 Sir G. Greenhill on Pendulum 



It will be seen that fused silica has qualities which commend! 

 it for use as a material for standards of length. A silica 

 standard metre is on the point of completion at the National 

 Physical Laboratory, and there is good reason to believe 

 that its adoption will be attended with success. 



The National Physical Laboratory, 

 TeddiiiRton. 



LXXXI. Pendulum Motion and Spherical Trigonometry, 

 By G. Greenhill *. 



MR. ROSE-INNES has developed the relation between 

 the revolution of a pendulum in a plane and the pro- 

 jection of the motion on a spherical surface, and he shows 

 that the argument of the elliptic function required can be 

 represented by an area on the sphere which grows uniformly 

 with the time (Phil. Mag. June 1910). 



In a change to the polar reciprocal, the time will then be 

 represented by a spherical arc, as discussed here in § 9. 



1. Consider a circle AQD on the vertical diameter AD, 

 and a particle Q circulating round it under gravity with 

 velocity due to the depth KQ below a horizontal line HK \, 

 the motion of Q will represent a pendulum making complete 

 revolutions, like a bicycle-wheel on its ball-bearings, put out 

 of balance by an iron bar in the spokes (fig. 1). 



The lettering and notation is that employed in my ' Elliptic 

 Functions,' fig. 13, where, with ADQ = </>, 



(1) KQ = AE-AN = AE-ADsin 2 <£ 



A D 



= AE(l--K 2 sin 2 </>) = AE.A 2 </>, * 2 = ^' 



(2) (vel. of Q) 2 = (AD ^)' = % • KP 



C d6 nt / (j 



J A^=k = "' * = * mU > n = VAO ; 



(3) 



so that njir is the number of beats per second in small oscil- 

 lation; and the elliptic argument u grows uniformly with the 

 time t, starting from the lowest point A. 



Draw the circle, centre E and radius EB, orthogonal to 



* Communicated bv the Author. 



