88 Preliminary Propositions 



charged : the quantity of redundant fluid in each of them will be to that in the 



third cylinder in a ratio between that of log ^ to log ^ + log T^T- and 



2EC . 2EC EC + Eb 



that of log -_- to log y, R + log ^r , provided the redundant fluid in 



the third cylinder is disposed in the same manner as in the other two. 



C JS 



JO 



Fig. 4. 



For let us suppose that ADFB and adfb are connected to the third cylinder 

 by the canal GM, then, if the redundant fluid in each cylinder is disposed 

 uniformly on its convex surface, the sum of the repulsions of ADFB and adfb 

 on the canal gM will be to the repulsion of the third cylinder thereon (supposing 

 the quantity of redundant fluid in it to be equal to that in each of the two others) 

 2EG . , EG + Eg ,_ ,__.2EG 



as log 7^- + log - 



to log 



"Gg b CB 



Let us now suppose the fluid in the first two cylinders to be disposed so as 

 to be in equilibrio, and consequently to be disposed in greater quantity near 

 their extremities than near their middles, and let the fluid in the third cylinder 

 be disposed in the same manner, and be the same in quantity as before. The 

 repulsion of ADFB on Gg will be diminished in a greater ratio, and consequently 

 its repulsion on gM will be diminished in a less ratio than that of adfb on gM, 

 consequently the sum of the repulsions of ADFB and adfb on gM will be 

 diminished in a less ratio than that of the third cylinder thereon, and therefore 

 the sum of the repulsions of ADFB and adfb on gM will be to that of the third 

 cylinder thereon in a greater ratio than that of 



2.EG EG 



log - + log 



Eg ^ 2EG 

 to log - . 



Therefore the real quantity of redundant fluid in each of the first two 

 cylinders will be to that in the third cylinder in a less ratio than that of 

 2EG 2EG EG + Eg 



In like manner, by supposing them to be connected to the third cylinder 

 by the canal bD, it may be shewn that the quantity of redundant fluid in either 

 of the first two cylinders is to that in the third in a greater ratio than that of 



2EC . 2EC EC + Eb* 



log -rrr to log -j^- + log 



CB 



CB 



Cb 



[* Note 13, p. 388.] 



