684 Proceedings of the Royal Society of Edinburgh. [Sess. 
a 
b c 
a 
a 
a!' 
a' 
V c 
ft 
ft' 
ft" 
a" 
b" c" 
y 
/ 
7 
y" 
In the second place, by substituting for n x only there is obtained 
aax + aa'y + aa'z + bu 2 + cu 3 = v l ' 
a ax 4 - aa'y + aa'z + b'u 2 + cu 3 = v 2 
a" ax + a" ay + a!'o!'z + b"u 2 + c"u 3 = v 3 
f3x + fly + /3"z - u 2 = 0 
yx+ y'y± y"z - u % = 0 . , 
whence comes for x an expression whose denominator is 
aa aa! aa!' b c 
a a a! a a a" b ’ c 
a" a a!' a! a!' a!' b" c" 
P P P' ~ 1 • 
y i y" - 1 
A comparison of the two denominators is supposed to establish the desired 
result ; but, although the dropping of the two negative units in the live- 
line determinant is quite justifiable, no allusion is made to it. 
It may be added that Sylvester’s umbral notation is used throughout 
in dealing with the subjects just referred to, 
( 11 ) ( 12 ) ... ( 1 ») 
j 1 2 ... n ) 
II 2 ... n) 
or 
( 21 ) ( 22 ) 
(nl) (n2) 
being used for one of the two determinants, and 
(2 n) 
(nn) 
p: 2 : • 
Ir 2 ' . 
. . n ) 
' > or 
. . n ) 
(ny 
(21)' 
(12)' .. 
(22)' .. 
. {In)' 
• <a»y 
(»1)' 
(»2)' .. 
. (nn)' 
for the other. The reading is thus rendered tiresome, and inaccurate 
printing exaggerates the trouble. 
Grassmann [H.] (1854, February, April). 
[Sur les differents genres de multiplication. Crelle’s Journal , xlix. 
pp. 123-141.] 
[Extrait d’un memoire de M. Grassmann. Comptes rendus . . . 
Acad, des Sci. (Paris), xxxviii. pp. 743-744.] 
Grassmann, having become aware of Cauchy’s three communications to 
the French Academy in January of 1853, claims that the principles there 
