764 
Proceedings of Royal Society of Edinburgh. [sess. 
The conviction that the identity ought to come out as a result of 
elimination, and the ingenious fulfilment of it by using the identity 
a + /? + y = 0 after the manner of Sylvester’s paper of 1840 are very 
noteworthy. 
It is finally noticed that “ the additional equation that exists 
between the distances of five points on a sphere ” can be similarly 
obtained, and the process is given. 
GRUNERT (1842). 
[LTeber die Theorie der Elimination. Archiv der Math. u. Phys., 
ii. pp. 76-105, 345-377.] 
This paper, extending to more than sixty pages, is little else 
than an amplified reproduction of work by Cauchy. Nine pages 
at the beginning concern simultaneous linear equations; the rest is 
entirely taken up with the various modes of eliminating x between 
two algebraical equations, <£(#) = 0, \]/(x) = 0. 
In the former part, which seems based on the third chapter of the 
Cours d’ Analyse, the only fresh matter is a lengthy proof of the 
proposition that the difference-product of any number of quantities 
changes sign when two of the quantities are transposed. It will 
suffice to note in regard to it that the so-called inductive method 
is followed, and that two cases have to be considered, viz. (1) when 
the new quantity is not one of the two which are interchanged, (2) 
when it is. (hi. 38) 
The second part follows closely Cauchy’s memoir of 1840. 
TERQUEM (1842). 
[Notice sur l’elimination. Formules de Cramer. Nouv. Annates de 
Math., i. pp. 125-131.*] 
This is merely a simply written exposition of Cramer’s rule, 
and of Bezout’s rule of 1779, and contains nothing noteworthy. 
It is curious, however, to observe the reason given for directing 
attention to Cramer’s rule, — “ Comme ce procede ne se trouve 
* The continuation intimated at the close (p. 131) was never made. 
