23© 



Report S.A.A. Advancement of Science. 



and thus arrived at the equation 

 Ae- 2. 1 biC2 1 . 1 &2C3 1 • ! h^i ! = 



^hhcs I 



whence, on account of the determinant on the extreme right being 

 equal to | &1C2I. | Vsl- 1 ^s^'il , it followed that 



Ae = j'hVsi^- 



3. About seven and a half years later the same special case 

 was utilized by A. Scholtz, of Buda-Pest, in his paper " Sechs 

 Punkte eines Kegelschnittes " {Airhiv d. Math. 11. Phys. LXII., pp. 

 317-324, year 1878). Scholtz gave no reference to Brill, of whose 

 paper he was most probably unaware, but established the result for 

 himself. The first row of the determinant he modified by 

 multiplying each element of it by jVsl and adding to the 

 product {a^b^l times the corresponding element of the fifth 

 row and |f2^3| times the corresponding element of the sixth 

 row ; the second and third rows were modified in like manner. 

 This, on putting 



D = l^iVsl, 

 led to 



As- I &2C3 I • I ^2^3 I • I «2^3 I = 



and all that remained was to show that the three-line determinant 

 on the right is equal to — \ 63^3 1 . | ^■2^3 1 . | ^2^3 1 • 



4. Strictly speaking the difference between the two proofs is 

 less even than might at first appear, for the modification effected by 

 Scholtz upon the first three rows can be accomplished exactly after 

 Brill's fashion, viz., by multiplying column- wise by 



V3I 



I «2&3 I 

 I '-"2«3 I 



C2«3 I 



I «2^3 ! 



; ^2^3 1 



i «2&3 I 

 I ^2'^3 I 

 I &2C3 1 



