92 Preliminary Propositions 



shall be very small in respect of the distance of C from the nearest part of the 

 circumference of AB, and let the least radius of curvature of the surface of AB 

 be so great in respect of CE that a point R may be taken such that CR shall 

 be small in respect of that radius of curvature, and yet very great in respect 

 of CE. 



Let Pp be a flat circular plate whose center is G and whose plane is per- 

 pendicular to GZ, and let its area be equal to that of AB, and let the quantity 

 of matter in it be also equal to that in AB, and let it be disposed uniformly: 

 the sum of the repulsions of AB and DF on CE in the opposite directions CE 

 and EC will be to the repulsion of Pp on the infinite column GZ very nearly 

 as 2CE to GP. 



For suppose each particle of matter in all that part of AB whose distance 

 from C is not greater than CR and in the corresponding part of DF to be 

 transferred to its corresponding point in Tt, so as to form a circular plate whose 

 radius is CR. 



If we suppose thJt the thickness of the plates Tt and Pp are both equal to 

 that of AB, the matter in all parts of Tt will be very nearly twice as dense as 

 that in AB or as that in Pp. Therefore the repulsion of Tt on CE will be very 

 nearly twice the repulsion of Pp on Gg, supposing Gg to be equal to CE. 



But from the foregoing lemma it appears that the sum of the repulsions 

 which the above-mentioned part of AB and DF exerted on CE before the 

 matter was transferred is very nearly equal to that which Tt exerts thereon 

 after the matter is transferred, and the sum of the repulsions of the remaining 

 part of AB and DF, or that whose distance from C is greater than CR, is very 

 small in respect of that part whose distance is less, therefore the sum of the 

 repulsions of the whole plates AB and DF on CE is to the repulsion of Pp 

 on GZ very nearly as 2.CE to GP. 



It may perhaps be supposed from this demonstration that it would be 

 necessary that CE should be excessively small in respect of CV, in order that 

 the sum of the repulsions of the plates on CE should be very nearly equal to 

 the repulsion of Pp on Gg, but in reality this seems not to be the case, for if 

 the plates are segments of concentric spheres whose center is V, the sum of 

 their repulsions will exceed twice the repulsion of Pp on Gg in a not much 



CE 

 greater ratio than that of i + ^. to i, and if the radius of curvature of their 



surfaces is in some places greater than CV, and nowhere less, I should think 

 that the sum of their repulsion could hardly exceed twice the repulsion of Pp 

 in so great a ratio as that. 



157] COR. II. If we now suppose that the matter of the plate AB is denser 

 near the circumference than near the point C, and that the density at and 

 near C is to the mean density (or the density which it would everywhere be of 

 if the matter was spread uniformly) as S to one, and that the quantity of matter 

 in each part of DF is equal to that in the corresponding part of AB as before, 

 the sum of the repulsions of the plates on CE will be less than if the matter 



