of the Stability of Dr. A. IV. Stewart's Atom. 261 



the forces called into play by an infinitesimal displacement 

 and see whether the direction of the force is such as to cause 

 the displaced electron to return to its original position. 

 This must be done for radial and tangential displacements in 

 the plane of the rings and for displacements perpendicular 

 to the plane of the rings. 



Condition? for Stability. 



Radial Displacements. 



Suppose an electron to be displaced radially so that its 

 distance from the centre of the orbit becomes a + da. 

 The force called into play 



(2-7T \ (2lT 

 Y 6 ) + 6d 2 cos 2 \ - +d> 



9^3 "*" 2f "~ [ . 9.77." \ i 3 



< a 2 + d 2 — 2ad cos ( - - -f (/> I r 

 2a 2 — tiarcos(~? + 0\ 4 6r 2 cos 2 (~ +0\ 



| a 2 + r 2_9 ar C0S ^ + ^T 



For radial stability, the quantity inside the bracket (taking 

 into account the proper sign for each term in the summations) 

 must be positive. 



Tangential Displacements. 



Suppose an electron is displaced so that the radius through 

 it turns through an angle dyjr. 



Then the tangential displacement is ad-yfr and the force 

 called into play 



For tangential stability the quantity inside the bracket 

 must be positive. 



