336 



The equations (6) and (7) therefore have the same meaning, as 

 soon as we have 



P^êj_ ^ J" A. ^ 



P ^ 



_ — (8) 



and it is only when these relations hold that either of the equations 

 (6) and (7) is a direct consequence, of the Other. 

 To simplify somewhat the notation I put 



M-rN 



(? = 



N-^L 



V = 



L+M 



M-N N—L ' L—M 



the new constants «, i?, y being related by the equation 



(1 + «)(!+ .i) 1 + y) + (1 - «) (1 ^) (1 - r) = 

 or by 



1 + /?r + r« + «i^ = , . . . . 



and consequently instead of (2) we may write 



P.ny 



(1 + y) 5, - ^^ + (!-«) ^. , 



(9) 



(10) 



(11) 



P,V',=(l-^)§: + (l+«) 1,-^3. ^ 



Comparing now the two sets of equations (8) and (11), we observe 

 that (11) defines a hornographic transformation expressing the quan- 

 tities Pip in the quantities §, and that the inverse transformation is 

 given by (&). 



Writing down the determinant 



- 1 1 - y 1 + /? 



1 + y — 1 1 — « 



1 — ^ 1 + « — 1 



of the first transformation and also the determinant 



a h-irK 9^9i 



of the 



p p 



9-\-9i 

 P P 



inverse transformation. 



P P 



b 



P.' 



f-A 

 P.P. 



in 



P^P. 



f±A 

 p p 



c 



p? 



which 



I have interchanged 



lines and columns, a known proposition says that the elements 

 of the latter determinant are proportional to the corresponding minors 

 of the former. Nine ratios therefore are equal to one and the same 



