Charge on circular plate 49 



the matter therein, it follows that the pressure against AB will be very 

 little more than on the first supposition. 



N.B. If any part of the cylinder is undercharged, the pressure against 

 AB is greater than the force with which the bodies attract. If the electric 

 repulsion is inversely as the square or some higher power of the distance, 

 it seems very unlikely that any part of the cylinder should be under- 

 charged; but if the repulsion is inversely as some lower power than the 

 square, it is not improbable but some part of the cylinder may be under- 

 charged. 



55] LEMMA VII. Let AB (Fig. 10) represent an infinitely thin flat 

 circular plate, seen edgeways, so as to appear 

 to the eye as a straight line; let C be the center 

 of the circle; and let DC passing through C, 

 be perpendicular to the plane of the plate ; and 

 let the plate be of uniform thickness, and con- 

 sist of uniform matter, whose particles repel 

 with a force inversely as the n power of the 

 distance; n being greater than one, and less 

 than three: the repulsion of the plate on a Fig. 10. 



particle at D is proportional to 



DC DC 



DC" 1 - 1 ~ DA"- 1 ' 



provided the thickness of the plate and size of the particle D is given. 



For if CA is supposed to flow, the corresponding fluxion of the quantity 

 of matter in the plate is proportional to CA x CA; and the corresponding 

 fluxion of the repulsion of the plate on the particle D, in the direction DC, 

 is proportional to 



CA x CA DC _ DA x DC . 



DA n X DA ' DA n 



for DA : CA :: CA : DA; the variable part of the fluent of which is 



'-f^--i- whence the repulsion of the plate on the particle D is 



DC DC DC DC 



proportional to ( ^^ - ^---^-^ , o o ^ n - 1 - 



56] COR. If DC"- 1 is very small in respect of CA"- 1 , the particle D 

 is repelled with very nearly the same force as if the diameter of the plate 

 was infinite. 



57] LEMMA VIII. Let L and / represent the two legs of a right-angled 

 triangle, and h the hypothenuse ; if the shorter leg / is so much less than 

 the other, that I"- 1 is very small in respect of L"- 1 , h 3 ~ n - L 3 ~ n will be 

 very small in respect of I 3 ~ n . 



C.P. i. 4 



