ELECTRICITY. 



531 



Theoretical as another, or it will not be impelled in any direction. 



Electricity. This i s demonstrated in Xrn't. Priiicip. lib. i. prop. 70. 



l"emmal Lemma 5. If the repulsion is inversely as the square 



PLATE f tne distance, a particle placed any where without the 



CCH. sphere BDE, is repelled by that sphere, and also by the 



Fig. 2. space BA, with the same force that it would if all the 



matter therein was collected in the centre of the sphere ; 



provided the density of the matter in it is every where 



the same at the same distance from the centre. This is 



easily deduced from Prop. 71. of the same book, and has 



been demonstrated by other authors. 



Pnnr. 5. Prop. V. Prob. 1. Let the sphere BDE be filled with 



t'ig. 3. uniform solid matter, overcharged with electric fluid ; 

 let the fluid in it be moveable, but unable to escape from 

 it ; let the fluid in the rest of infinite space be moveable, 

 and sufficient to saturate the matter in it ; and let the 

 matter in the whole of infinite space, or at least in the 

 space B/3, whose dimensions will be given below, be 

 uniform and solid ; and let the law of the electric at- 

 traction and repulsion be inversely as the square of the 

 distance ; it is required to determine in what manner the 

 fluid will be disposed both within and without the globe. 

 Take the space B6 such, that the interstices between 

 the particles of matter in it shall be just sufficient to 

 hold a quantity of electric fluid, whose particles are 

 pressed close together, so as to touch each other, equal 

 to the whole redundant fluid in the globe, besides the 

 quantity requisite to saturate the matter in BA; and 

 take the space B.3 such, that the matter in it shall be 

 just able to saturate the redundant fluid in the globe ; 

 then, in all parts of the space Kb, the fluid will be 

 pressed close together, so that its particles shall touch 

 each other ; the space K,3 will be entirely deprived of 

 fluid; and in the space Cb, and all the rest of infinite 

 pace, the matter will be exactly saturated. 



For, if the fluid is disposed in the above mentioned 

 manner, a particle of fluid placed any where within the 

 space C b will not be impelled in any direction by the 

 fluid in B b, or the matter in Bfl, and will therefore have 

 no tendency to move. A particle placed any where 

 without the sphere /33i will be attracted with just as 

 much force by the matter in K/3, as it is repelled by the 

 redundant fluid in BA, and will therefore have no ten- 

 dency to move. A particle placed any where within 

 the space B.3, will indeed be repelled towards the sur- 

 face, by all the redundant fluid in that space which is 

 placed nearer the centre than itself; but as the fluid in 

 that space is already pressed as close together as pos- 

 sible, it will not have any tendency to move ; and in 

 the space H,3 there is no fluid to move, so that no part 

 of the fluid can have any tendency to move. 



Besides, it seems impossible for the fluid to be at 

 rent, if it is disposed in any other fonn ; for if the den- 

 ity of the fluid is not every where the same at the same 

 distance from the centre, but is greater near b than near 

 ff, a particle placed any where between these two points 

 will move from 6 towards d ; but if the density is every 

 where the same at the same distance from the centre, 

 and the fluid in B A is not pressed close together, the 

 space CA will l>e overcharged, and consequently a par- 

 ticle at A will l>e repelled from the centre, and cannot 

 be at rest. In like manner, if there is any fluid in 1! 3, 

 it cannot l>e at rest. And, by the same kind of reason- 

 ing, it might be shown, that, if the fluid is not spread 

 uniformly within the space Cb, and without the sphere 

 4)i, it cannot be at rest. 



Carol. 1. If the globe BDE is undercharged, every Theoreticr.) 

 thing else being the same as before, there will be a Electricity.^ 

 space B b, in which the matter will be entirely deprived p^^T""' 

 of fluid, and a space B,3, in which the fluid will be pres- CCLI. 

 sed close together; the matter in BA being equal to the Fig. 2. 

 whole redundant matter in the globe, and the redun- 

 dant fluid in B, being just sufficient to saturate the 

 matter in B b ; and in all the rest of space the matter 

 will be exactly saturated. The demonstration is exactly 

 similar to the foregoing. 



Coral. 2. The fluid in the globe BDE will be dis- 

 posed in exactly the same manner, whether the fluid 

 without is immoveable, and disposed in such a man- 

 ner that the matter shall be every where saturated, or 

 whether it is disposed as above described ; and the 

 fluid without the globe will be disposed in just the 

 same manner, whether the fluid within is disposed uni- 

 formly, or whether it is disposed as above described. 



Lemma 6. Let the whole space comprehended be- Lemma 8. 

 tween two parallel planes, infinitely extended each 

 way, be filled with uniform matter, the repulsion of 

 whose particles is inversely as the square of the dis- 

 tance ; the plate of matter formed thereby will repel 

 a particle of matter with exactly the same force, at 

 whatever distance from it, it be placed. 



For, suppose that there are two such plates, of equal 

 thickness, placed parallel to each other, let A ( Fig. 3. ) Fig. 3. 

 be any point not placed in or between the two plates : 

 let BCD represent any part of the nearest plate : draw 

 the lines AB, AC, and AD, cutting the farthest plate 

 in b, c, and d ; for it is plain, that if they cut one plate, 

 they must, if producefl, cut the other : the triangle 

 BCD is to the triangle be d, as AB* to A A 1 ; therefore, 

 a particle of matter at A will be repelled with the same 

 force by the matter in the triangle BCD, as by that in 

 bed. Whence it appears, that a particle at A will be re- 

 pelled with as much force by the nearest plate, as by 

 the more distant ; and consequently will be impelled 

 with the same force by either plate, at whatever dis- 

 tance from it, it be placed. 



Prop. VI. Prob. 2. In Fig. 4. let the parallel lines Aa, p H8P . s 

 BA, &c. represent parallel planes infinitely extended [,. 4. 

 each way : let the spaces* AD and EH be filled with 

 uniform solid matter : let the electric fluid in each of 

 those spaces be moveable and unable to escape ; and let 

 all the rest of the matter in the universe be saturated 

 with immoveable fluid ; and let the electric attraction 

 and repulsion be inversely as the square of the distance. 

 It is required to determine in what manner the fluid 

 will be disposed in the spaces AD and EH, according 

 as one or lx>th of them are over and under charged. 



Let AD be that space which contains the greatest 

 quantity of redundant fluid, if both spaces are over- 

 charged, or which contains the least redundant matter, 

 if both are undercharged; or, if one is overcharged, and 

 the other undercharged, let AD be the overcharged 

 one. Then, first, there will be two spaces, AB and 

 GH, which will either be entirely deprived of fluid, or 

 in which the particles will be pressed close together; 

 namely, if the whole quantity of fluid in AD and EH 

 together, is less than sufficient to saturate the matter 

 therein, they will be entirely deprived of fluid, the 

 quantity of redundant matter in each being half the 

 whole redundant matter in AD and EH together; but 

 if the fluid in AD and EH together is more than suffi- 

 cient to saturate the matter, the fluid in AB and OH 



By the ipace AD or AB, &r. is meant the pc comprehended between Uie planes Aa and Drf, or bttwten Ai nd JU. 



