to Five Points in a Plane. 



463 



The equations of the line OA', or say of the line 0(41, 23), are 

 those of the planes 41, 23, viz. these are 



x 



y 



=o, 



x, y , z 

 *, + 1, 7 



=0; 



« + l, > 7 

 a + 1, /3-f 1, 7 + 1 

 that is, 



*(/8-y)- (« + l)(y-*)=0, 

 a?(/8 + y + l)-a (y + *) = 0. 

 "Writing for shortness 



M=«+/3 + y + l, 

 these equations give 



_ 2«(« + l) 1_ 1 



* : y : *~ (M + 27a) (M.+ 2*0) : M + 2y* : M + 2*0' 



or completing the system, 

 for line OA' we have 



_ 3«(« + l) 1 _L . 



v.y.z- (M + 27«)(M + 2a/3) : M + 27a : M + 2«^ 



for line OB' we have 



_1 2/3Q8 + 1) 1 



57 : P-*- M + 2/3y ' (M + 2a/3)(M + 2/3y) ' M + 2a/3' 

 for line OC we have 



1 1 t 2 7 (y + l) 



x :y:z: 



M + 2^7*M + 27« *(M + 2^7)(M + 2ya) 



The equations of the lines 0*, 00, O7 are of course (y = 0, z = 0), 

 (z=Q, x=0), (x=0, y = 0) respectively; and we therefore see at 

 once that the planes (OA', 0*), (OB', 00), (OC, O7) meet in a 

 ine, viz. in the line which has for its equations 



1 



1 



* : y :z= M + 2(3y : M+2y* : M + 2*~p' 



The lines OB', OC, 0* will lie in a plane, if only 



4/3 7 (/3 + l)(y + l) . 

 i_ (M + 2/3y) 2 ' 



that is, 



(M+2/3y) 2 =4/3y(/3 + l)(y+l), 



or, as this may be written, 



M 2 + 4 / 87(a+/3+7=l+/37)=4^7(^7 + ^+7 + l); "' 



