RATIONAL INTEGRAL CONTINUED FRACTION, 
537 
we may pass to the actual case from any case where the roots are infinitesimally 
near to the actual roots of F(^), and all infinitesimally different from one another. 
Moreover the choice of the infinitesimal variations being arbitrary, let the h, roots 
c, be replaced by a group of roots 
where is a prime root of the equation = and ^ is an infinitesimal quantity, and 
suppose each of the other groups to be varied in an analogous manner. Then it 
may easily be shown from this that the one of the formulae in question will become 
V 
Fc, 
and similarly, the twin formula becomes 
* 
^7e 
Corresponding modifications will admit of being made by aid of a like method in 
the general formulae of art. (^.) upon a similar supposition as to equalities springing 
up between the roots of fx per se and of <p{x) per se, or between the I’oots offx and 
<px inter se. 
Art. (n.). If in (art. 1.) we take i=0, the formula for Q._n will become 
Q. = K„., B 
X 
hf h, ...hf 
X 
_^tu + l ^tu + % 
^ j). 
u and i> being any two integers whose sum is ly,, which is identical (as it ought to be) 
with the expi'ession virtually contained in the formulae of Section II. for the syzygetic 
multiplier of 0(a?) in the syzygetic equation connecting Fa: and Oa: with their fii-st 
residue when Oa: is supposed to be dimensions in x lower than Fa: identical, 
videlicet, in other words, with the integer part of the algebraical fraction 
For in general if |9 is a prime root of the equation p" = l, and if fx have w roots all equal to c and 4'^ is 
any other function of x and if $ is an infinitesimal quantity, then rejecting all powers of $ higher than the 
(w — l)th degree, 
4,(0+$) ^ 4,(0+ pS) ^ \l(c + p^S) , , 4 ^(c + p'^-'S) 
/'(c + s) f(c+ pS) V(c + p~S ) ■ /'(c+r-'J) 
■■ ^ r 1 
1 
(d\ 
Kdc) 
1 d/cwJ"'-! 1 
( d'\ 
\dc) 
W— 1 
1 4'C 
1 
<0 
1 fc 
4 A 2 
