C. Barns — Repulsion of Ttvo Metallic Disks. 95 



The gravitational attraction/* of the disks necessarily involves 

 spherical harmonics, but may be written temporarily as 



f'=ymm'/f(d) 



where m ; is the mass of the stationary disk at a mean distance 

 d from m. Equating these forces and inserting the value of 

 i^ R , the equation for AJV, the displacement at the micrometer, 

 becomes 



y m' 2 R 2 m/M 



AN = 



f(d)g<j>h{\ -hmR/Mh) 



In the first place, therefore, the lightest available pendulum 

 and the heaviest admissible disk is to be selected, although the 

 increase of sensitiveness is not quite proportional to m/M. 

 This procedure, even when the float* is used, is relatively 

 inefficient. 



It is much more effective, since the disks are all but in con- 

 tact on measurement, to increase their area and decrease their 

 thickness. Equation 3, if equal disks of radius r and thick- 

 ness t and density p are employed, and if/* (d) — d 2 = f takes 

 the form 



AN ^'^ 



V ~ g <j>hM(l + mR/Mh 



so that for disks and pendulum of an invariable mass, AN 

 increases roughly as the fourth power of the radius of the 

 disks. Inasmuch as this is also the case in which the gravita- 

 tional attraction may be much more easily computed, it is, 

 therefore, the direction which the experiment should take. 

 Thus, if the above disks for constant mass be decreased in area 

 and increased in thickness and be placed all but in contact, the 

 estimated value of AN should be at least 



r == 



5 



10 



15 



r'= 



7-5 



15 



22*5 etc. 



10'AiV= 



1-2 



18-7 



94-8 



quantities easily measurable even without the use of an inter- 

 ferometer. 



3. Observations. — The brass plates of the preliminary work 

 were soon replaced by a set, larger in area but thinner, this 

 being in the direction of the improvement of method indicated. 

 The same unnecessarily heavy steel pendulum had, however, 

 to be used, so that M = 1250 grams, h = 80 cm , R = lll'3 cm , 



*The equations given for the float apply only if the float is placed at the 

 center of gravity of the pendulum. If the float supports the axis of the 

 pendulum, the original equations are unchanged, the effect being merely to 

 damp the pendulum and to take the weight off the pivots. 



