386 
Proceedings of Royal Society of Edinburgh. [sess. 
and by using Newton’s relations 
na n = a n s 0 , 
(n-l)a n ^ = a n s L + a n _ 1 s 0 , 
(n — 2)a n _ 2 = ct n s 2 + a n _ 1 s 1 + ci, n _ 2 s 0 , 
(n — 3)a n _ s = (X m S 3 + + & n _ 2 S l + ®n- 3 S 0 > 
the elements of the last n rows of the right-hand determinant, we 
are told, can be so changed that in each there will occur only one 
of the a ! s and that in the first power. Thereupon the conclusion 
is formally announced that the -determinant with which we 
started can be expressed as a rational integral function of the 
(’2n- 2) th degree in the quantities 
«o ^n— 1 
" 5 5 * ' * J 5 
a n a n a n 
and that the said function becomes homogeneous on multiplication 
by The actual result is not given, but in the second 
edition (1864) it is stated to be 
a n 
«n-l 
a n _ 2 . . . . 
0 
«n- 1 - - - - 
0 
0 
a n .... 
0 
na n 
- - - - 
na n 
(n - 2)a n _ 2 . . . . 
a n - 1 
2a n _ 2 
3a n _ s . . . . 
“eine Determinante (2n - 2)ten Grades, bei welcher die m-2 
ersten und die m — 1 folgenden Zeilen in Bezug auf die nicht 
verschwindenden Elemente ubereinstimmen.” 
Part of the object which Baltzer had here in view was to 
establish the relation between two forms of the discriminant of 
the given equation ; namely, that obtained by squaring the 
determinant-form of the difference-product and that obtained as 
the eliminant of the equations 
f\x) = 0 , nf(x) - xf\x) fi 0 , 
or the equations 
(a n x n + a n _{x n y + . . . +a Q y n ) = 0 
