Repulsion by columns 85 



For supposing CK to flow, the fluxion of its repulsion on B in the direction 

 CB is proportional to ^rg- 2 x VR> ^ e fluent f which is ,^ p rr> . an d is 



nothing when CK is nothing. 



148] LEMMA XV. Fig. 3. Let GEFHMN be a cylinder whose bases are 

 GEF and HMN and whose axis is CK. Let the convex surface of this cylinder 

 be uniformly coated with matter, and let GC be small in respect of CK. Let 



Fig. 3- 



GA be a diameter of the base produced, and D any point therein. The repulsion 

 of the convex surface of the cylinder on the point D in the direction CD is very 

 nearly the same as if all the matter therein was collected in the axis CK and 

 spread uniformly therein. 



For let MED and meD be two planes infinitely near to each other, parallel 

 to CK and passing through D, and cutting the convex surface in ME and NF 

 and in me and nf, which will consequently be right lines equal to each other 

 and perpendicular to ED; and draw CP perpendicular to ED. 



The repulsion of NnfF on D in the direction CD is proportional to 



Ff x FN PD Ee x EM PD 

 p D ;un x rff ancl tliat * Mme E ls proportional to ^ ^^ x - 



Ff Ee 



But Ff is to Ee as FD to ED, therefore ^^ and v^. are each equal to 



^=- = nf : , therefore the sum of the repulsions of MmeE and NnfF 

 ru + L.L) 2rL) 



is proportional to 



(Ff+Ee)CKxPD / i _i_\ _ (Ff + Ee)CK /_i_ _i_\ 

 2.PD x CD \ND + MD) 2CD \ND """ MDj ' 



But the repulsion of the same quantity of matter collected in CK is proportional 



(Ff+ Ee) x CK 

 - ~ - 



to 



2 , I 



x -~~. , and, as CG is small in respect of CK, + 



