1\S 'POSITIONS BE81I 



VII. 1: 'no of two forces be pivcn in 



and also M of their resultant. tin- '. 



of tl.< "tln-r 



.11 be a straight li: 



Let a and r denote the magnitudes of t w 

 the former to make an angle a with the dim-lion of the 

 resultant, and the latter an angle 6. 



:i. rcsolvini: al. mi; the straight line which is at right 

 angles to : '.^ resuh. ..ave 



a sin a r sin = 0. 



Now a and a being given, while r and are variaMo, this 

 tion represents a straight line which is parallel to the, 

 :ion of the resultant, and at a distance a sin a from it. 

 See Conic Sections, Chap. n. 



VIII. From any point within a re-ular polygon ]: 

 dieulars are drawn on all the sides of tl. : . that 



iirection of tin- n-ultant of all the forces represented 



iiculars passes through the f the 



circle circumscribing the polygon, and find the le of 

 the resultant. 



Let p denote the perpendicular from the centre on a 



c t lie distance of the point at which the forces act from the 



, a the an^le which this distance makes with a fixed 



M itli th.- perpendicular from the 



< on a side, n the number of sides in the polygon ; and let 



aJ 



P* 



Then the magnitude of the w" 1 force may be d< 

 J9 CC08 (m/9 a), and the direction of this force will ; 

 an angle mft witli the fixed straight line. 



:ice the resolved parts of the forces parallel t<> the : 

 straight line, and at right angles to it, will be respectiv* 



2 [p - c cos (m{3 a)} cos mfi and 2 [p - c cos (mp a) ] sin m/3, 



! denotes a summation to be taken with respect to m 

 from m=l to m = n. 



