138 Proceedings of Royal Society of Edinburgh. [sess. 
Finally, and to more purpose, it is noted that if we pass from 
, a 2 , . . . , a t ) to the readily-suggested extension 
m 1 Zj 
^1 ^2 ^2 
w 2 Z 3 
»<-! ™>i li 
the corresponding fundamental theorem is 
h ^i+j 
1 ) m 2 J • • * i m i+j+ 1 
. . n 
i+j 
( h I'i—l \ / ^i+1 ^+i 
m 1} m 2 , . . . , m i V m i+1 , m* +2 , . . . , m i+j+1 
n \ n i- 1 ' ' W t+1 n i+j 
l\ li - 2 
li+2 
i+j 
- IgiA m 1 , ra 2 , . . . , jf ra i+2 , ra* +3 , . . . , m i+i+1 
' ^£-2 ' ' ^i+2 
Sylvester, J. J. (1853, Oct., JSTov.). 
[On a theory of the syzygetic relations of two rational integral 
functions, comprising an application to the theory of 
Sturm’s functions, and that of the greatest algebraical 
common measure. Phil . Trans . Roy. Soc. London , cxliii. 
pp. 407-548.] 
Although this lengthy memoir in its original form bears date 
“ 16th June 1853,” it is the equally lengthy “supplements” added 
later while passing through the press that claim attention in the 
present connection. In the first of these (§ i., p. 474) the de- 
nominator of the fraction 
q n 
is denoted by [g 1} q 2 , . . . , gj, and termed a “cumulant,” and 
throughout the later portion of the paper this name constantly 
recurs. It is not, however, until we come to the second 
