LXXI. On Induced Stability. By Andrew Stephenson *. 



1. 1VTOT only may a single pivoted body be maintained 

 X. \ in the position of unstable equilibrium by vertical 

 oscillation of the pivot, but furthermore a number of freely 

 jointed links may be similarly maintained. 



In the simplest case, when there are only two links, the 

 conditions of stability are readily found. With the usual 

 notation the equations of motion in the neighbourhood of the 

 vertical are 



!(// 1 2 + // 1 2 + ^ 1 2 )D 2 -( / y + 2«/rcos/^( /i i + ^ / i)}^ + ^2l) 2 ^ = 0, 

 { (V + *i f ) I> 2 — {<J + 2ara 2 cos nt)h 2 }<t> + / 1 AoD 2 6>= 0, 



where fju is the ratio of the mass of the second, or upper, rod 



* An addition to a paper under the above title, Feb. 1908, Communi- 

 cated by the Author. 





On Induced Stability. 765 



evolution of the radium-emanation could not be assumed to 

 be subordinate in amount to thorium-emanation. However, 

 it is to be supposed that the much shorter-lived emanation of 

 thorium would be more local in its effects than the radium- 

 emanation, which, breaking up slowly, has time to be diffused 

 over a wider area. 



The amount of thorium in sea-water is small, and until an 

 observation was made on a large quantity of water, only 

 negative results were obtained. A thorium salt added to 

 sea- water gives an immediate precipitate (which is cleared 

 by an acid). It may be that the poverty in thorium of the 

 oceanic waters and the underlying sediments is not a true 

 indication of the thorium content of the rocks, but is to be 

 referred to chemical effects consigning the thorium to the 

 littoral deposits. 



The geological bearing of the distribution of thorium cannot 

 be discussed without a knowledge of the rate of transforma- 

 tion of that element. According to Bragg (Phil. Mag. June 

 1906) this element- is breaking up at one-fifth the rate of 

 uranium. If this is so, and taking into account the lesser 

 emission of a particles by the derivatives of thorium, the 

 indications of the results given above assign to thorium a 

 relatively unimportant position compared with uranium as a 

 heat-producing agent in the rocks. On the other hand, if 

 the rates of transformation are about equal (Soddy, Phil. Mag. 

 Oct. 1908), the importance of thorium might be comparable 

 with that of uranium, the quantities of the two elements in 

 the rocks not appearing to differ greatly in amount. 



