584 MR THOMAS STEPHENS DA VIES ON 



if from, any point whatever Z lines Z P 1? Z P 2 , . . . . Z P m 6e drawn perpendicu- 

 lar to the m, given lines, and ZP to the line found, we shall always have 



Let the given lines be referred to any line perpendicular to them, as polar 

 axis ; and let the origin be the controi'd of the points in which the axis cuts the 

 given lines. Denote the distances of these respective points from the centroid by 

 p v p 2 , . . . . p m \ and "hyp, the distance of the porismatic line from the centroid. 

 Also, let r 6 denote the arbitrary point Z. 



Then the equations of the given lines will be 



p 1 = r cos 0, p 2 = r cos 6, . . . . p m = r cos 6. 



And (see BUTTON'S Course, ii. p. 268, 12th ed.) the perpendiculars will be ex- 

 pressed by 



Z P x = db ( Pl - r cos 0), Z P 2 = (p 2 - r cos 0) ; etc. 



Wherefore the equation of the porism becomes 



a i (PI ~ r cos ^) 2 + ^2 (PI ~ r cos ^) 2 + Sa i(P ~ r cos ^) 2 + * 2 } ; 

 or expanding, cancelling, and equating the co-efficients of cos 6 to 0, we have simply 

 p. Sa m = S(a m p m ) ......... (1) 



S(a mPm ) = Sa m . (/ + *) ....... (2) 



Now, since the origin is the centroid, we have from (1) 



S (a m p m ) = ; and hence p = 0, . . . . (3) 



or the line sought passes through the centroid. 



Again, from (2) and (3) we get the porismatic space 



Sa 



PROPOSITIONS XV., XIX. PORISMS. 



Let there be given m lines all meeting in one point, and m magnitudes 

 j, 2 , . . . , a m : then there can be found tno other lines, also passing through 

 the same point, such, that if from any point Z there be drawn perpendicu- 

 lars Z P p Z P 2 . . . . , Z P m , to the given lines, and likewise Z Q 15 Z Qa, to those 

 found, tee shall always have 



2S(a m Z PJ) = Sa m .{ZQ 1 * + Z Q/}. 



Let the points in which all the lines meet be taken as polar origin, the 

 axis being any whatever. Let the angles made by the perpendiculars to the 

 given lines with the axis be 6 lt &, . . . , & m , and those made by the perpendiculars 

 to the porismatic lines be w^ w 2 : then if r 6 be the arbitrary point Z, we shall have 



Z P x = =fc r cos (0 - 6J, Z P 2 = r cos (6 - 2 ), etc. 

 Z Qj = r cos (6 - wj, Z Q 2 = r cos (Q - ,). 



