112 Proceedings of Royal Society of Edinburgh. [sess. 
S fabcd). The result forced upon him, however, would be the 
single term 
£_ 4 PD(0 abed ) , 
which in modern notation is 
|a 2 6W| . 
In the course of the work, therefore, the term | a 1 & 3 c 4 d 6 | would be 
dropped altogether out of sight. The cause of this is undoubtedly 
the imposition of the condition just mentioned ; — indeed, if we 
take the result of the work as above performed in the modern 
notation, viz. : — 
|aWd 6 | - 3|aWd»| } 
and make the elements periodic, i.e ., make 
a 6 ,6 6 ,c 6 ,# = a 1 ,^ 1 ,^ 1 ,^ 1 , 
the first alternant will vanish by reason of having two indices alike, 
and we shall he left with a result agreeing with Sylvester’s. 
The conclusion, therefore, which we are tempted to draw is that 
if Sylvester’s general theorem be correct it is only when the 
elements are subjected to periodicity. 
JACOBI (1841). 
[De functionibus alternantibus earumque divisione per productum 
e differentiis elementorum conflatum. Crellds Journ., xxii. 
pp. 360-371.] 
After having treated of determinants in general (pp. 285-318), 
and of the special form which afterwards came to bear his own 
name (pp. 319-359), Jacobi turned to another special form which 
he had learned about from his great predecessor Cauchy. As, 
however, he differed from Cauchy in his mode of defining a 
determinant, Cauchy’s definition, which, it will he remembered, 
