53 i 



ELECTRICITY. 



! I.-* '.n.itjr 



rt-T 



Km. 9. 



l.'emnu 7. 



- 



to the plane AB. in which the fluid will be 

 dote together ; and the fluid in that space will 

 P*** *8"*?* t ^ P ll n* AB, and will endeavour to es- 

 cape from it ; and, by Prop. 2. the two Ixxlies w ill nt- 

 **"*teach other : then the furce with which the Hurl 

 pratw* against the plane AB, is very nearly the same 



ith which the two bodies attract each other in the di- 

 rection EA, provided that no part of AEFB is under- 

 charged. 



Suppose to much of the fluid in each part of the cy- 

 linder, a* it sufficient to saturate the matter in that part, 

 to become solid, the remainder, or the redundant fluid, 

 r; innining fluid as before. In this case, the 



' * *"*<*} vi it in (*; 1 1 c 



___ : the plane AB must be exactly equal to that w ith 

 vhich the two bodies attract each other in the direction 

 I-.A : for the force with which D attracts that part of 

 the fluid which we supposed to become solid, is exactly 

 equal to that with which it repels the matter in the cy- 

 linder; and the redundant fluid in Ea6F is at liberty 

 to move, if it had any tendency to do so, without mov- 

 ing the cylinder ; so that the only thing which has any 

 tendency to impel the cylinder in the direction EA, is 

 the pressure of the redundant fluid in AaAB against 

 All ; and as the part near EF is saturated, there is no 

 redundant fluid to press against the plane EF, and thus 

 to counteract the pressure against AB. Suppose m,w 

 the electric fluid in the cylinder to become fluid, 

 the force with which the two bodies attract each other 

 will remain exactly the same; and the only alteration 

 m the pressure against AB will be, that that part of the 

 fluid in AaAB, which we at first supposed solid and 

 unable to press against the plane, will now be at liberty 

 to press against it ; but as the density of the fluid, w lien 

 its particles are pmaed close together, may be suppo- 

 sed many times greater than when it is no denser than 

 sufficient to saturate the matter in the cylinder, and 

 consequently the quantity of redundant fluid in AJB 

 many times greater than that which is required to sa- 

 turate the matter in it, it follows that the pressure 

 against AB will be very little more than on the first 

 supposition. 



\ H. 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 dis- 

 tance, it seems very unlikely that any part of the cylin- 

 'der should be undercharged. 



Ismma 7. Let AB (Fig. 10. ) represent an infinitely 

 turn. at circular plate, seen edgewise, so as to appear 

 to the eye as a straight line ; let C be the centre of the 

 circle ; and let DC, passing through C, be perpendicu- 

 lar to the plane of the plate; and let the plate be of 

 uniform thickness, and consist of uniform matter, whose 

 particles rej>el with a force inversely as the n power of 

 the distance; n being greater than 1, and less than 3 : 

 the repulsion of the plate on a particle at D, is Dronor- 



DC DC 



' DC^^DA 5 ^' P 1 " 1 " 1 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 pro- 

 portional 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 _^ v *^. 

 DAxDC DA " ^' 



DA. 



for DA is to CA : : CA : DA, the vari- 



DC 



-.able part of the fluent of which is 



whence the repulsion of the plate on the particle D, i* Tllc 



pr o p ortional t Electricity. 



IK DC . DC_ 



"F^DXF*' 



**L PtATr 



W I X DC * ri i ^ Ul\ - U\* U/\ * I.I.I. 



"'. It DC- -i is very small in respect of CA"-', K ' s "' 

 the particle D is repelled with very nearly the same 

 t'liive as it' the diameter of the plate was infinite. 



/ nma 8. LetL and / represent the two legs of aright 1-emmi f. 

 angled triangle, and h the hypothcnuse; if the shorter 

 leg / is so much less than the other, that /* ' is very 

 small in respect of L" ' , then A'-" L J ~" will be very 

 small in respect of/'". 



For /,'-"= 



= LI X 1 + 



3 wx/ 1 S-nxn l 



2L 



8L 



2L*- 8L"<" 



&c. which i.i very small in respect of/'", as /" ' is by 

 the supposition very small in respect of L*- 1 . 



1. :r,ma 9. Let DC now represent the axis of a cylin- Lcranu t. 

 dric or prismatic column of uniform matter ; and let 

 the diameter of the column be so small, that the repul- 

 sion of the plate AB on it shall not be sensibly different 

 from what it would be, if all the matter in it was col- 

 lected in the axis. The force with which the plate re- 

 pels the column, is proportional to DC 5 -" + AC'- 

 DA'-", supposing the thickness of the plate and base 

 of the column to be given. For if DC is supposed to 

 flow, the corresponding fluxion of the repulsion is pro- 



AC'-" -f. DC' D A'-" 



,vanishe> 



the fluent of which, 



when DC vanishes. 



Carol. 1. If the length of the column is so great, tha* 

 AC"', is very small in respect of DC"', the repulsion 

 of the plate on it is very nearly the same as if the co- 

 lumn was infinitely continued. For, by lemma 8, 

 AC^-f-DC 3 -" DA'~" differs very little in this case 

 from AC'-" ; and if DC is infinite, it is exactly equal 

 to it. 



Coral. 2. If AC"- 1 is very small in respect of DC"-', 

 and the point E be taken in DC, such thnt EC"-' shall 

 be very small in respect of AC" 1 , the repulsion of the 

 plate on the small part of the column EC, is to its rr ; .ul- 

 sion on the whole column DC, very nearly as EC'~" " to 

 AC'-". 



Lemma 10. If we now suppose all the matter of the Lemra* i*. 

 plate to be collected in the circumference of the circle, 

 so as to form an infinitely slender uniform ring, its re- 

 pulsion on the column DC will be less than when the 

 matter is spread uniformly all over the plate in the ra- 



* 1 X DA*- 1 



AC-"_ DA'-". 



For it was before said, that if the matter of the plate 

 be spread uniformly, its repulsion on the column will 

 be proportional to DC' " + AC' " DA'-", or may be 

 expressed by it ; let now AC, the semidiameter of the 

 plate, be increased by the infinitely small quantity AC ; 

 the quantity of matter in the plate will be increased by 

 a quantity, which is to the whole aa 2 AC to AC ; and 

 the repujlsion of the plate on the column will be increa- 



