1 



272 FACTION. CIRCUL.U: I. \MI\\. 



If OP and OQ be produced to meet the li 

 P and Q respectively, it may be shewn that 

 <>f th,- rl, Q oo a particle at is equal tn that of PQ, 



and iu this manner wr prove what we have just shewn, 



that the attractions of .I// ami J '/.'' >n a particle at are 

 equal and coincident. This proposition is given in Karnshaw's 

 Dynamics, p. 326. 



It easily follows, that if a particle ! attracted l.y the i 

 sides of a triangle, it will be in equilibrium if it be ; 

 the centre of the circle inscribed in the triangle. 



207. To find the a 1 of a /?/ 



on a particle situated in a straty/it Jim <lrn,m thwnjh the 

 centre of the lamina at right angles to its plane. 



Suppose C the centre of the circle DAB, the plane of the 

 paper coinciding with one face of 

 the lamina, and the attracted par- 

 ticle being in a straight line drawn 

 through C perpendicular to the 

 lamina and at a distance c from 

 C. Describe from the centre C 

 t\vo adjacent concentric circles, one 

 with radius CPr, and the other 

 with radius CQ = r + Sr. Let tc 

 denote the thickness of the lamina. 

 which is supposed to be an in- 

 definitely small quantity, then the mass of the circular 

 contained between the adjacent circles is ZTrpicrbr. Every 

 particle in this circular ring is at a distance V(c*-f r f ) li"in 

 the attracted particle; also the resultant attraction of th- 

 is in the straight line throi < | right angles to the lamina, 

 and is equal to 



the factor ... t . being the multiplier necessary in order to 



resolve the attraction of any element of the ring along the 

 normal to the lamina through C. 



