586 MR THOMAS STEPHENS DAVIES ON 



PROPOSITIONS XVI., XX. PORISMS. 



Let there be given m lines, which are neither all parallel nor all meet in one point, 

 and m magnitudes, a lt a? . . . . , a m : then there can be found two other straight 

 lines and a space s\ such, that if perpendiculars Z P 19 Z P 2 , . . . . Z P m be 

 drawn to all the given lines, and others Z Q t , Z Q to those found, we shall 

 always have 



2S(a m Z P m *) = S m 



For, let the several given lines be 



Pi = r cos (0-0J, />, = ; cos(0-0 2 ), ---- ,p m =rcos (0-0 ra ); 



and let the porismatic lines be denoted by 



q L = r cos (6 Wj), and y 2 = r cos (6 2 ). 



Then the perpendiculars upon these from the arbitrary point Z, or r 6, will 

 be (HuTTON, ii. p. 268), 



(p l ~rcos(6~e i )}, {j 2 -r cos (0-0 2 )}, etc.; and 

 {0! rcos(0 Wj)}, and {? 2 cos(0-w 2 )}. 



Insert these values in the equation of the porism, and arrange the results in 

 terms r cos 0, r cos 2 0, r sin 0, and r sin 20; then, as r 2 cancels, we have 



2S(a mPm 2 ) = Sa m . {tf + gf + s 2 } .......... (1) 



2 S(a m p m cos m ) = Sa m . [q l cos o^+y., cos w 2 } ....... (2) 



2 S(a m p m sin m ) = S a m . {^ sin ! + ? 2 sin W 3 } ....... (3) 



2 5 (a m cos 2 m ) = 5 1 a m . {cos 2 Wj + cos 2 W 2 } ....... (4) 



2 S (a m sin 2 m ) = S a m . {sin 2 Wj+sin 2 W 2 } ........ (5) 



From (4, 5) will be found, as in the preceding porism, the values of Wj w a "; 

 from these results, combined with (2, 3), we shall obtain, by a simple equation, 

 the values of q^qi\ and from these values inserted in (1) we finally obtain the 

 value of the space s 2 . 



PROPOSITIONS XVII., XXI. PORISMS. 



Let there be given m lines, which are neither all parallel nor all meet in one point, 

 and m magnitudes, a l5 a*, . . . . , a m : then there can be found three lines, such, 

 drawing from any point, Z, the perpendiculars Z P l? Z P 2 , . . . . , Z P m to the 

 given lines, and Z Qi, Z Q 2 , Z Q } to those found, we shall always have 



. (Z Q 2 ). 



