LIQUIDS AND ALLIED EXPERIMENTS. 43 



Since equation (2) is linear in I and equation (1) quadratic in /,the curves 

 of f e and/ m in terms of / must in general intersect in two points, one of which 

 (lower or smaller /) corresponds to stable and the other (upper or larger /) 

 to unstable equilibrium of the disk. For a fixed value of D these are 

 further apart as V is smaller. When V increases sufficiently the two points 

 of intersection eventually coalesce in a single point. This particular value 

 of I shows the highest stable position which the disk may reach. For large 

 values of V there is no point of intersection, or the disk passes without 

 interruption from the lower to the upper plate of the condenser. The same 

 result may be brought about by decreasing D for a fixed V, and on this 

 principle I have based the following method of measurement. 



If we differentiate equations (1) and (2) with respect to I the results are 



d M = ill U) 



dl 2.262 Xio\D-l) 3 K * J 



Tl~R 2pg (5) 



If these slopes are identical 



F 2 =; 2.262 Xio 6 - 2 pg{D-lf (6) 



2 K 



Now, when there is but a single point of intersection (tangency of equations 

 (1) and (2)), equations (3) and (6) correspond to the same value of l = l c , 

 whence after canceling superfluous quantities 



D=3l e (7) 



In other words, if the electrical forces are sufficiently strong to raise the 

 disk e more than one-third of the distance D between the plates of the 

 condenser, it will pass all the way to the upper disk cc. Hence under these 

 circumstances equation (3) gives 



V 2 = o. 5 Q2 7 Xio 6 r 2 D 2 /R 2 (8) 



Thus in case of the first method of measurement the upper plate cc, fig. 1 1 a, 

 is to be gradually lowered, while the disk e rises, until the last position of 

 stable equilibrium is just exceeded, or the disk travels to the upper plate. 

 The guard ring may now be raised D/3 and a closer adjustment made. 



29. Constants of the Tubular Float. — To show the numerical relations 

 involved, the values of f m and f e may be computed and represented graph- 

 ically in terms of the lift I. Since for water p=i, and the diameters of 

 stem and disk are 0.854 cm - an d 6.65 cm., respectively, 



o 



r 



f m =-jf ps 1 = i6 - lSl d y nes 



which is the oblique line through the origin of the graph in fig. 14. 



