30 Ellery IV. Davis 



k becomes infinite, the red vector becomes tangent to the black 

 circle, becomes, in fact, either / or /. The Q's belong to the 

 circular rays ; all the other red elements belong to the circle. 

 Perhaps a better way of treating this case is the following : 

 The red vectors collinear with the origin and connecting the 

 black circle 



X * + y*= 3X ' a +y'* = a ,Z 



to the blue circle 



x 2 +y 2 = (x' + x"Y + (/ + y"Y= (a' + a") 2 , 



evidently satisfy the equation. Let us look upon this double 

 circle as the fundamental curve. 

 Then 



x 2 — y 2 = .\\'"~ — y / 2 = a' 2 

 and 



-r 2 -/ = (-r/ + V) 2 + (y/ + y/') 2 = (a' + a") 2 , 



form together the supplementary curve .x\ 2 — y 1 2 = a 2 . That is 

 to say, a curve such that, if the ordinates be multiplied by i, leav- 

 ing the abscissas unchanged, the resulting double-vectors satisfy 



x 2 -\- y 2 = a 2 . 

 This gives 



/= — 3'i"- y"=yi- 



Now just as vectors radial from the origin satisfy the fun- 

 damental curve so do such vectors satisfy the supplementary 

 curve. And just as on the fundamental curve 



x" y" _ a" 

 x T= y' == a r ' 



so also on the supplementary does 



x'" v." a" 



x' }'/ a' 



Therefore 



(.r' f /)EE«, -y(')^[x^-^y^ 



30 



