Comparison of charges 87 



the same proportion to that in the globe in whatever part the canal meets the 

 cylinder, therefore first I say the redundant fluid in the cylinder will bear a 



C ' K 



greater proportion to that in the globe than that of - _ to nat. log. 



For let the canal TR be straight and perpendicular to BL, and let it meet 

 the cylinder in R, the middle point of the line BL, and let it, if produced, meet 

 the axis in S, which will consequently be the middle point of CK; then, if the 

 redundant fluid in the cylinder was spread uniformly on its convex surface, 

 the quantity of redundant fluid therein would be to that in the globe very nearly 



CK . . . CK 

 as to nat. log. . 



For in that case the repulsion of the cylinder on the canal RT would be to 

 the repulsion of the same quantity of redundant fluid collected in C very nearly 



2SK . SK CK . CK 



as nat. log. -=^- to = or as nat. log. -^ to ^ , and the force with which 

 o/v jJK Co 20/3 



the globe repels the canal in the direction TR is the same with which a quantity 

 of redundant fluid equal to that in the globe placed at S would repel it in the 

 contrary direction. 



But there can be no doubt but that almost all the redundant fluid in the 

 cylinder will be collected on its surface, and also will be collected in greater 

 quantity near the ends than near the middle, consequently the repulsion of the 

 cylinder on RT will be less than if the redundant fluid was spread uniformly 

 on its convex surface, and therefore the quantity of redundant fluid in it will 

 bear a greater proportion to that in the globe than it would on that supposition. 



Secondly, the quantity of fluid in the cylinder will bear a less proportion 



to that in the globe than that of -^-^ to nat. log. _ 



For suppose the canal to meet the cylinder in B and to coincide with BA . 

 Then, if the redundant fluid was spread uniformly on the convex surface, the 



C 'K 

 quantity therein would be to that in the globe very nearly as -^-= to nat. log. 



O/5 



-7^-5- , and the real quantity of redundant fluid in it will bear a less proportion 

 CD 



to that in the globe than if it was spread uniformly on the convex surface. 



151] COR. Therefore the quantity of redundant fluid in the cylinder is to 

 that in a globe whose diameter equals CK in a ratio between that of 2 to nat. 



2CK CK* 



log. -^o- and that of i to nat. log. -^ . 



152] PROP. XXXII. Fig. 4. Let ADFB and adfb be two equal cylinders 

 whose axes are EC and ec, let them be parallel to each other and placed so that 

 Cc, the line joining the ends of the axes, shall be perpendicular to the axes, 

 and let the lines EC and fb be bisected in G and g, and let them be connected 

 by canals of incompressible fluid of any shape to a third cylinder of the same 

 size and shape placed at an infinite distance from them, and let them be over- 



[* Note 12, p. 382.] 



