g 6 Preliminary Propositions 



By the supposition the force with which AB repels the canal EM is M m, 

 and the force with which the same quantity of redundant fluid, spread on DF 

 in the same manner in which the deficient fluid therein is actually disposed, 

 repels it is G, therefore if the redundant fluid in A B is called A, the deficient 



fluid in DF will be A x ~ - ; therefore the force with which DF attracts 



Cr 



CG is (G g) T*~ an d the excess of the force with which AB repels CG 



above that with which DF attracts it is 



(G-g) (M-m) _ Mg+Gm-mg 



G G 



which must be equal to the force with which a quantity of fluid equal to B 



T> 



spread uniformly over Ww would repel it, that is, it must be equal to W j ; 



BGW 

 therefore A equals Mg+Gm _ mg - 



161] COR. I. If the plate of glass is flat, and its thickness is very small in 

 respect of the least distance of the point C from the circumference of AB, and 

 the fluid in AB and DF is spread uniformly, the quantity of redundant fluid 



B x CW 

 in DF will differ very little from ^g , and the deficient fluid in DF will 



be very nearly equal to the redundant fluid in AB. 



For as the plate of glass is flat, the two coatings will be equal to each other, 

 and therefore M and G are equal to each other, and so are m and g, and 



CE* 

 s.. differs very little from prp- , and moreover g is very small in respect of G. 



162] COR. II. If the plate is flat and the two coatings are circular, their 

 centers being in C and E, the quantity of redundant fluid in AB will be more 



B x CW CW 



accurately equal to - g^ x cw _ c j~ , CV being in this case equal to the 



semi-diameter of the coatings, and the deficient fluid in DF will be to the 

 redundant in AB nearly as CW - CE to CW. 



For in this case ^ is accurately equal to 



CW-VCE*+CW* 



CW 



and therefore _ 



2m _ m? 2CEVCW*+CE*-2CE* 

 W W z ~ CW* 



which, if CE is small in respect of CW, differs very little from 



2CE (CW - CE) 

 CW> 



* Lemma XV [Art. 148.] 



