410 Proceedings of Royal Society of Edinburgh. [sess. 
branch of the subject that identities between two zeta-ic multipli- 
cations of difference-products really belong. 
This early paper, one cannot but observe, has all the characteristics 
afterwards so familiar to readers of Sylvester’s writings, — fervid 
imagination, vigorous originality, bold exuberance of diction, hasty 
if not contemptuous disregard of historical research, the outstrip- 
ping of demonstration by enunciation, and an infective enthusiasm 
as to the vistas opened up by his work. 
SYLVESTER (1840). 
[A method of determining by mere inspection the derivatives 
from two equations of any degree. Philosophical Magazine , 
xvi. pp. 132-135.] 
The two equations taken are 
a n x n + + . . . + ape + a 0 = 0 ) 
b n x n + b n _ itf" -1 + . . . + bpc + 6 0 = 0 \ f 
and rules are given for attaining three different objects, viz. (1) a 
rule for absolutely eliminating x; (2) a rule for finding the prime 
derivative of the first degree, that is to say of the form Ax - B = 0 ; 
(3) a rule for finding the prime derivative of any degree. The first 
of these concerns the process afterwards so well known by the name 
“ dialytic.” Only part of it need be given (p. 132): — 
“ Eorm out of the a progression of coefficients m lines, and 
in like manner out of the b progression of coefficients form n 
lines in the following manner : Attach m — 1 zeros all to the 
right of the terms in the a progression ; next attach m - 2 
zeros to the right and carry 1 over to the left ; next attach 
m - 3 zeros to the right and carry 2 over to the left. Proceed 
in like manner until all the m - 1 zeros are carried over to the 
left, and none remain on the right. The m lines thus formed 
are to be written under one another. 
Proceed in like manner to form n lines out of the b pro- 
gression by scattering n - 1 zeros between the right and left. 
If we write these n lines under the m lines last obtained, we 
