Mr. T. S. Davies on Pascal's Mystic Hexagram. 41 



the locus (or loci) of E and F. A more independent and per- 

 haps more elegant process, would be the following ; the ge- 

 neral principle, however, being the same as that before em- 

 ployed. 



Let the absolute lengths 

 of the lines O A, O B, O C, 

 O D be a, /3, y, 8; then the 

 several points concerned will 

 be denoted as under. 



(A)....(0,-«) 

 (B)....(-/3,0) 

 (C)....(y,0) 



(D).... (0,3) 



(F)....(^) 

 (E) ,...(a? 3 y 2 ) 



and the several lines concerned will be expressed in the usual 

 manner, thus : — 



(BD).... -/3y + 8*=-/38 (1.) 



(AC) .... yy — *%= — ay (2.) 



(AF) .... *!(y+«)=*(y, + «) (3.) 



(BE) .... y(ar 9 + |3)=y a (* + jB). . . (4.) 



(CF) .... t,(x l -y) = y 1 (x-y) (5.) 



(DE) .... * 9 (y-8) = *(y 8 -8) (6.) 



Denoting as before G and H by (Xj Y a ) and (X 2 Y 2 ), we get 



(«g a + /3y 2 + «£)*! 



*i #2 - (*« + 0) (yi + a ) 



(8 x x + y y, - y 8) # a 



X 1= - 



x 2 = + 



Y,= - 



Y,= + 



#a 2/1 - to - y) (y« - 8 ) 



(q^t + $y x + «/3)y 2 



*i 2/2 - (*« + Z 3 ) (yi + «) 



(Sjt 2 + 7^ 2 -7 8 )y 2 



> 



(70 



* 2 yi- (*i -7) (y 2 - 8 ). 



Also, since (^j y x ) is in (1.), and (ar 8 y^ in (2.), we have the 

 equations, 



/3yi = ^ x + /38 (8.) 



yy 2 = «.r 2 -ay (9.) 



In' the values of X 15 X 2 substitute the values ofy x , y 2 from 

 (8.) and (9.), and in those of Y x , Y 2 those of x v x^ from the 



