84 Preliminary Propositions 



municate with EF by separate canals, and whether the canals by which they 

 communicate with it are straight or crooked, or in whatever direction EF is 

 placed in respect of them, provided the situation of AB and ab in respect of 

 each other remains the same. Only it must be observed that if the fluid in the 

 plates is not disposed so as to be in equilibrio, as will most likely be the case 

 if it is disposed as in the two preceding corollaries, it is necessary that the canals 

 should meet them in their centers, for if the fluid in a plate is not in equilibrio, 

 its repulsion on a canal of infinite length will not be the same in whatever part 

 the canal meets it, as it will if the fluid in the plate is in equilibrio. 



145] LEMMA XII. Fig. 2. Let BA be an infinitely slender cylindric column 

 of uniform matter infinitely continued beyond A : the repulsion of a particle of 



C 



Fig. 2. 



matter K on this column in the direction BA is proportional to or may be 

 represented by -^ supposing the size of the particle and [the] base of the 

 column to be given. 



For draw KC perpendicular to AB continued, and let the point B flow 

 towards C, the fluxion of the repulsion of K on the column equals 



-OT CB _ - KB' 

 KB* X KB ~ KB* ' 



the fluent of which, ^5 , is nothing when KB is infinite. 

 ti.ts 



146] LEMMA XIII. Suppose now KC to represent an infinitely slender 

 cylindric column of uniform matter: the repulsion of KC on the infinite column 

 BA is to the repulsion of the same quantity of matter collected in the point C 



t KC + KB^ KC 

 on the same column as the nat. log. of ^r^ to -^ . 



For the repulsion of all the matter therein, when collected at C, on BA is 



KC 

 proportional to prp , and supposing the column CK to flow, the fluxion of its 



O.D 



CK' KC -L Kfl 

 repulsion on BA is equal to T*W, the fluent of which is the nat. log. of - ^= , 



and is nothing when CK is nothing. 



147] LEMMA XIV. The repulsion of CK on a particle at B, in the direction 



CK 

 CB, is proportional to -& ^ , supposing the base of CK and the size of 



f\LJ X ' / ' 



the particle B to be given. 



