588 MR THOMAS STEPHENS DAVIES ON 



Now, as there are ten equations, all independent of each other, it follows 

 that jp=5 ; for there will be five gs and five ws to be determined from this sys- 

 tem, and these are the requisite conditions for finding five lines. Whence the 

 number of lines is incorrectly given by Dr STEWART who porismatises only four 

 lines ; but the Porism is evidently possible with the condition altered as here pro- 

 posed. 



PROPOSITIONS XXX., XXXI. PORISMS. 



Let there be given m points Aj, A 2 , . . . . A m , and m magnitudes, a lt 2 , . a m : 

 then there can be found tno lines OX, Y, and a point P, together tcith 

 two magnitudes a. b ; such, that if from any point whaterer, Z, there be drawn 

 lines to all the given points, and to the point found, together with perpendicu- 

 lars Z X, Z Y to the lines found, me shall always haw 



S (a m . A m Z*) = S a m . {P Z< + o(Z X 2 + Z Y 2 + 6') }. 



Let, as before, the given points be referred to the centroi'd as origin of polar 

 co-ordinates, and denoted by r l 6^ r z 2 , r m , 6 m ; also denote P by r # > Z by r 6, 

 and the lines O X, Y by 



/> 1 = r cos (6 wj 



p z r co (6 W 2 ) 



Then the perpendiculars on these from r B will be expresed as before by 



Z X^rit^-r cos (0-wJ} 

 Z Y=db{j0 2 -r cos (0- Wo)} 



Also, the lines from Z to the several points will be expressed in power by 

 A M Z< = r< + 4 rr w + r w -4r r m (r m * + r>) cos (6-6^ + 2 r m * r* cos 2 (fl-fl.) 

 P Z 4 = r 1 + 4 r 2 r 2 + r* - 4 r r (r 2 + r 2 ) cos (6- ) + 2 r a 2 r 2 cos 2 (0- ) 



ZY 2 = (/; 2 + r 2 ) 







With these values form the equation of the porism, cancel common terms, 

 and equate to zero the co- efficients of the arbitrary quantities which remain; 

 then we get the following system of conditional equations : 



