164 



Dr. G. W. Royston-Pigott on 



[June 19, 



Aodr\ A 3 / 



where v is tlie " velocity of mean square " of (A.-^). 



When the distance of the spheres is so large that we may neglect 

 twelfth and higher inverse powers of the distance this attraction is 



18v 2 



+ 1 ' \r)a\\r^bV 2^+1 J ' 



2/a + 1 \rjal\r*— W 2 P - 



a, b, being the radii of the vibrating and free sphere respectively. 

 From this it may be shown that if the density of the sphere is greater 

 than the fluid it is attracted, whilst if less it will be attracted or re- 

 pelled, according as its distance is less or greater than a certain critical 



distance, which is given by r— j-z ~T7= - ^ • P° r instance, 



if p='9, this distance =7*648, the radius of the free sphere. These 



formulae are obtained on the supposition that the spheres are not so 



dA dA 

 close that — 1 are too great to be neglected, for at contact — 1 are 

 dr dr 



infinite. When the sphere is held fast, the mean force required 



dA . 1 

 to do so is — v 2 — 1, which, when we neglect twelfth powers of — 



dr r 



= X weight of fluid displaced bv the vibrating sphere'; for 



g ( r 2_&2)4 & 1 J 5 i > 



example, for two oxygen atoms at 0° C, at a distance four times their 

 radius, the force is about 78 X ] 6 weight of fluid displaced by one 

 atom, thus while the force decreases indefinitely, the effective force 

 increases indefinitely. 



For two spheres (a=b=^ inch r=4&) vibrating through a distance 

 xo-th of an inch 256 times a second, in water, the force is equal to the 

 weight of 12 "8 milligrams. 



VI. i( Microscopical Researches in High Power Definition." By 

 G. W. Royston-Pigott, M.A., M.D. Cantab., F.R.S. Re- 

 ceived May 23, 1879. 



(Abstract.) 



In its general scope the paper is intended to deal with difficulties 

 in microscopic research, usually found insuperable — such, for instance, 

 as the invisibility of minute closely packed refracting spherules, exist- 

 ing in double rouleaux, or promiscuously aggregated, when their 

 individual diameter varies between the l-80,000th to the l-200,000th 

 of an inch. 



