138 
Proceedings of the Royal Society of Edinburgh. [Sess. 
He does not, however, note, as Beltrami did in a special case early in the 
same year, that by using another method of elimination, namely, Euler’s 
of 1748, the eliminant is found to be 
n(a 0 — y + ct]Q r + aff r + . . . + 
where 0 r is an -71 th root of 1 : and thus he fails to bring out the fact that 
Spottiswoode’s result of 1853 is nothing more nor less than the statement 
of the equality of those two eliminants.* Instead of this he performs the 
operation 
colj + 0 r col 2 + 0 ? 2 .co1 3 + 6 r n ~ l do\ n 
and so arrives at a practically equivalent result, namely, that the equation 
C (a 0 - y , a x , a 2 , . . . , a n _ x ) = 0 
is satisfied by putting 
y = OLq -f- CL-^Oy -j- CLc)0\ -f- . . . + ^ . 
Two other points worth noting are (1) his calling C(a 0 , a x , a 2 , . . . , a n _ x ) 
the norm of a 0 + a x B r + a 2 0l + . . . -fi a n _ x 0^~ l , in accordance with an exten- 
sion of a usage of Gauss’, and (2) his statement that of the n n terms got 
by working out the product 
H +- Ct x O r + CLcffi. + . . . + Ct n L\0™ ^ 
only the 1.2.3. . . n terms of the determinant remain. In regard to the 
former it has to be remarked that 6 r must then be restricted to stand for 
a 'primitive n th root of 1, and in regard to the latter that even some of the 
1.2.3 . . .n terms of the determinant do not remain. 
Baltzer, R. (1870). 
[Theorie und Anwendung der Determinanten .... 3 te ver- 
besserte Aufl. viii + 242 pp., Leipzig.] 
In addition to a few changes in phraseology this edition contains the 
fresh theorem that in every term of C(a 0 , a x , a 2 , . . . , a n _ x ) the sum of the 
suffixes is divisible by n. The proof is based on the fact that the suffix of 
any element (i , k) is either — i + k or n — i + k according as i is less f or 
greater than k ; for from this it follows that in the case of any term 
(l,r)(2,s)(3A) 
* See History , ii. pp. 369-370. 
t He means “ not greater.” 
