Bodies connected to two parallel disks 57 



deficient fluid is in DF; let P be to one as the force with which the plate 

 AB would repel the fluid in CE, if the canal ME was continued to C, is 

 to the force with which it would repel the fluid in CM ; and let the force 

 with which AB repels the fluid in CG, be to the force with which it would 

 repel it, if the redundant fluid in it was spread uniformly, as TT to i; and 

 let the force with which the body H repels the fluid in CG, be the same with 

 which a quantity of redundant fluid, which we will call B, spread uniformly 

 over AB, would repel it in the contrary direction. Then will the redundant 



D 



fluid in A B be equal to ^-- pa , and therefore, if P is very small, will 



T> 



be very nearly equal to =- ; and the deficient fluid in DF will be to the 



redundant fluid in AB, as i P to i, and therefore, if P is very small, 

 will be very nearly equal to the redundant fluid in AB. 



For it is plain, that the force with which AB repels the fluid in EM, 

 must be equal to that with which DF attracts it; for otherwise, some 

 fluid would run out of DF into L, or out of L into DF: for the same 

 reason, the excess of the -repulsion of AB on the fluid in CG, above the 

 attraction of FD thereon, must be equal to the force with which a quantity 

 of redundant fluid equal to B, spread uniformly over AB, would repel it, 



T) 



or it must be equal to that with which a quantity equal to - , spread in 



the manner in which the redundant fluid is actually spread in AB, would 

 repel it. By the supposition, the force with which AB repels the fluid 

 in EM, is to the force with which it would repel the fluid in CM, supposing 

 EM to be continued to C, as i P to i ; but the force with which any 

 quantity of fluid in AB would repel the fluid in CM, is the same with which 

 an equal quantity similarly disposed in DF, would repel the fluid in EM; 

 therefore the force with which the redundant fluid in AB repels the fluid 

 in EM, is to that with which an equal quantity similarly disposed in DF, 

 would repel it, as i P to i: therefore, if the redundant fluid in A B be 

 called A , the deficient fluid in DF must be A x i P : for the same 

 reason, the force with which DF attracts the fluid in CG, is to that with 

 which AB repels it, as A x i - P x i P, or A x (i P) 2 , to A ; there- 

 fore, the excess of the force with which AB repels CG above that with 

 which DF attracts it, is equal to that with which a quantity of redundant 

 fluid equal to A A x (i P) 2 , or A x (zP - P 2 ), spread over AB, in 

 the manner in which the redundant fluid therein is actually spread, would 



repel it : therefore A x (zP P 2 ) must be equal to - , or A must be equal 



7T 



B 



to 



9.P7T 



75] COR. I. If the density of the redundant fluid near the middle of 

 the plate AB, is less than the mean density, or the density which it would 



