592 MR THOMAS STEPHENS DAVIES ON 



Again, for the second case of Prop. 36, all the lines meet in one point ; and 

 this being the origin of co-ordinates, we shall have p l =p 2 = =p m = ; and hence 

 from (1,2) we have 5 t 1 =9 r 2 = =q p =0, and all the porismatic lines pass through 

 the origin. 



For this reason, also, only those of the subsequent equations which do not 

 involve q r g 2 , q p have an existence in this case. These are (9, 10, 13, 14) ; 

 and under these circumstances they become 



S a m . S(cos2 u p )=p . S(a m cos 2 OT ) (33) 



Sa m . S(sm 2 P )=/> . S (a m sin 2 6 m ) (34) 



Sa m . S(cos4 w p )=p. S (a m cos 4 m ) (35) 



Sa m . S(sin4 u p )=p . S (a m sin 4 m ) (36) 



These, again, givep=4 instead of jp=3, as the proposition is stated by Dr 

 STEWART. 



PROPOSITIONS XLIII,, XLIV. PORISMS. 



Lei there be given m points A 15 A 2 , . . . ., A m , and as many magnitudes^ 

 15 (7 2, . . , a m ; and let n be any number (subject to conditions to be hereafter 

 determined} : then there can be found p points B l5 B 2 , . . . ., B p , such, that if 

 from any point, Z, lines be drawn to all the given points, and likewise to the 

 points found, ice shall always have 



p.S(a m .A m 7? n ) = Sa m . S (B, Z 2n ). 



Let *i 6 V r 3 6,,, . . . ., r m 6 m be the given points ; 

 MjWpMjWj, . . . ., Up tti p the porismatic ones ; and 

 r &, the arbitrary point Z. Then, 



A m Z 2 =r 2 -2r r m cos (6-6 m ) + r m 2 ; 

 B p Z 2 =r*-2r u p cos (6-to p ) + u P 2 ; 



which are the general types of the squares of the lines to be raised to the nih 

 power. 



Let them be so raised by Lemma i. : then the terms in r- " cancel from the 

 equation being respectively 



p ( ai r'- n +a 2 r* n + ---- +a m r 2n ), and 



the latter being carried to p terms. 



In the next place, equate the co-efficients of the several terms in r^ rf cos v 6, 

 and j * sin v 6, for all values of /j, and v within the degrees expressed by the 



