DR MATTHEW STEWART'S GENERAL THEOREMS. 583 



PEOPOSITIONS XI., XII. POEISMS. 



Let there be given m points and m magnitudes, as in the preceding : then there may 

 be found a circle, and likewise a point, such, that drawing any line through the 

 point found to cut the circle in X and Y, and that Z be any point whatever in 

 the same plane, we shall always have 



S(k m Z*)=Sa m . (XZ' + YZ 2 ) 



For, let the given points, referred to the centro'id as origin, and any axis 

 whatever be r t 6 lt r 2 2 , . . . , r m Q m ; and denote the points X, Y by Ui u lt u 2 w 2 , and 

 the radius of the circle by p ; and let Z be r 6. 



Then, expressing the lines concerned in terms of these quantities, cancelling 

 Sa n . r* from both sides, and equating to zero the co-efficients of r, 



S(<t m r m ->)=Sa m (u* + ufi . . >. . . . (1) 



Mj COS (5 &J 1 ) + M 2 COS(0 W 2 ) = ..... (2) 



But since also is arbitrary, (2) becomes 



M t COS Wj + U-j, COS W 2 = ......... (3) 



MJ sin Wj + W2 sin W 2 = ......... (4) 



These two equations are satisfied by any two points in a line passing through 



the origin, and equidistant from it, on opposite sides; or M 1 =w 2> and 1 ='jr4-w s . 

 The point required to be found is hence the centro'id, and the circle has that 



point for its centre. 



It also follows, that t = % = g the radius of the circle; and hence from (1) 



we have 





Whence the circle is entirely determined. 



Dr STEWART porismatises, not a circle, but two points, X and Y, to be found. 

 The equations to which in such form the proposition gives rise, are precisely the 

 same as those above : wherefore there would be given only the three equations 

 (1, 2, 3) for the determination of four quantities u v 2, w lf w 2 , which obviously 

 leaves one of the quantities indeterminate. Dr SMALL notices, in another form, 

 this indetenninateness. 



[See also note on these.] 



PEOPOSITIONS XIII., XVIII. POEISMS. 



Let there be given m parallel lines and m magnitudes a v a s , . . . , a m : then there 

 . can be found another line parallel to these, and likewise a space /, such, that 



