544 Proceedings of Royal Society of Edinburgh. 
Fit ilia, eliminationis negotio rite institute 
0 = {a - x){a 4- x){a" + x){a'” + x) 
-{a-x)(a!-^xyd - {a - x){a" + x)c”c'' - {a-x){a"^ + x)c"*c"' 
- {a" + x){f” + x)b'}) - (a" + x){a + x)h''b" - {a' + x){a” + x)h''b"' 
+ 2c'c"c'\a -x) 2cb”b"'(a' + x) (lii.) 
+ 2dT'b\P + x) + 2d"bP\a"^-hx) 
+ bPdd + V'b"d'c" + V"b'"d''d'^ - 2db"c'c" - 2V'b'"c"c'’' - 2b'%'c'”c'. 
From the next paragraph we learn his sources of information, and 
infer that as yet Cauchy’s memoir was unknown to him. The first 
sentence is (p. 239) — 
“ Inter sedecim quantitates a, yS, etc. et sedecim, quae ex iis 
derivantur. A, A', etc. plurimse intercedunt relationes perele- 
gantes, quse cum analystis ex iis, quse Laplace, Vandermonde, 
in commentariis academiae Parisiensis A. 1772 p. ii.. Gauss in 
disquis. arithm. sectio V., J. Binet in vol. ix. diariorum instituti 
polytechnici Pariensis, aliique tradiderunt, satis notae sint, 
paucas tantum referam, quae casu nostro speciali ope aequa- 
tionum (iv) facile ex iis derivantur.” 
The third memoir is by far the most important to us. In the 
course of the investigation a new special form of determinants, 
afterwards so well known by the designation shew determinants, 
turns up ; and three pages are devoted to an examination of the 
final expanded form of it. This examination, we cannot, of course, 
now enter upon ; hut it is of importance to note that in it Jacobi 
takes the step of adopting the name determinant^ — a fact which 
doubtless was decisive of the fate of the word. The adoption thus 
made (although stated to he from Gauss), and the occurre-' ce of the 
words “ Horizontalreihen,” “ Yerticalreihen,” make it proDable that 
Cauchy’s memoir had now come to his notice. 
