DR MATTHEW STEWART'S GENERAL THEOREMS. 577 



= Sa m .{r* + *-*} ............ (1) 



(2) 



S(a m r m cos 6 m ) = S a m . r cos 6 ............. (3) 



S (a m r m sin 6 m ) = S a m . r sin ............. (4) 



S (a m r m 3 cos 0,) = S a m . {r 3 cos 6 + -a? Qoj cos o^ +p 2 cos w 2 )} . . (5) 



4 



' 5 (a m r m 3 sin TO = S a m . [r a 3 sin 6 + - a 2 (^ sin Wj +_p 2 sin W 2 )} . . (6) 

 S ( 'm 2 cos 2 m ) = S a m . {/- 2 cos 2 + 1 a 2 (cos 2 Wj + cos 2 W 2 )} . . (7) 

 5 ( r w sin 2 m ) = 5 1 a m . [r 2 sin 2 ^ + i a 2 (sin 2 Wj + sin 2 w 2 )} . . (8) 



Now, since the left sides of (3, 4) are zero, we find that r =0, and that is 

 indeterminate. The point P is, therefore, the origin of co-ordinates, or the cen- 

 troid of the given system itself. 



Insert this value of r in (1) : then we get 



Insert it in (7, 8) : then these become 



4 S (a m . r m 2 cos 2 $,) 

 v 



^ ' 



(10) 



4: S (a m r m 2 sin 2 6 m ) 

 =- C^| - ........ (11) 



from which w x and w 3 may be found as in (xv. xix) ; and which, as in that place, 

 are real. 



Put the value of r and the values of Wj, w 2 in (5, 6) : then we get the values 

 of PV p t . 



Finally, from the insertion of the values ofp v p 2 and r a in (2), we obtain the 

 value of b 2 . 



PROPOSITIONS XXXII., XXXIII. POIUSM. 



Let there be given m points A x , A 2 , . . . ., A m , and m magnitudes a^ a v . . . ., a m : 

 then there mny be found four points B 15 B, Bs, B 4 , swcA, #A# if from any point, 

 Z, we </jw/! ?i//eA' ?o a// the given points, and likewise lines to all the points found y 

 we shall always have 



4S(a m . A m Z<) = a m . S (B 4 Z 4 ). 

 VOL. XV. PART IV. 7 U 



