86 Preliminary Propositions 



2 * 

 differs very little from -&-=: , therefore the sum of the repulsions of MmeE and 



NnfF is very nearly the same as if all the matter in them was collected in CK, 

 and consequently the repulsion of the whole convex surface of the cylinder will 

 be very nearly the same as if all the matter in it was collected in CK. 



149] COR. Therefore if BA represents an infinitely thin cylindric column 

 of uniform matter infinitely extended beyond A, the repulsion of the convex 

 surface of the cylinder thereon in the direction BA is very nearly the same as 

 if all the matter therein was collected in CK, and therefore is to the repulsion 

 of the same quantity of matter collected in the point C thereon very nearly as 



CK + KB . CK zCK A CK " 



nat. log. ^rs -- to -^D> that is very nearly as nat. log. -^- to ^ In 



like manner the repulsion on the infinite column DA is to the repulsion of the 



4- KD CK 

 same quantity of matter collected in C very nearly as nat. log. ^== to ^ . 



150] PROP. XXXI. Fig. 3. Let the cylinder GEFHMN be connected to 

 the globe W, whose diameter is equal to GB and whose distance from it is 

 infinite, by a canal TR of incompressible fluid of any shape, and meeting the 

 cylinder in any part, and let them be overcharged: the quantity of redundant 

 fluid in the cylinder will be to that in the globe in a less ratio than that of CK 



to nat. log. -^=- , and in a greater ratio than that of -= to nat. log. -~5 



provided CB is small in respect of CK. 



By Prop. XXIV the quantity of redundant fluid in the cylinder will bear 

 * As neither MD nor ND differ from KD by so much as CB, it is plain that 



II 2 



j-fri + TFVJ cannot differ from -=^j= in so great a proportion as that of BC to KD, 



but in reality it does not differ from it in so great a ratio as that of CB 2 to KD 1 , 

 but as it is not material being so exact, I shall omit the demonstration. See A. I. 

 [From MS. " A. i "] Demonstration of note at bottom of page 8, 

 CB = r, CP = b, PF = d, PD = a, CR* + CD* = e a , 

 b* - d* = /, e 2 - / 2 = g*. 



*-*+ 



g e "*" e 3 ' 



ND* = CR* + a* - 2ad + d* = e- - b* - 2ad + d* = e* - f* - 2ad 

 - g 2 - 2ad, 

 g 2 + 2nd, 



I ad 



g + g 3 + 3 2g& _ 2 



I ad a*d* ~g 



g g 3 

 2 / aW _2 b* a*d* _ d* 



which is less than 



2 b* d* _ 2 



~~ 



e e 3 e* ~~ e e 3 



