416 
(tX 
MR. SYLVESTER ON THE RESIDUES OF ^ EXPANDED 
(5.) 
and in general we shall have 
R,=:Q^.Q+P,.P, 
where it is evident that will be of e 4 -(<— 1)5 ('“0 dimensions in x. 
Art (2.) Hence it follows that the ratios P : Q, : R, may be ascertained by the direct 
application of the method of indeterminate coefficients, for Q, will contain e+L and 
P will contain / disposable constants, making e+2/ disposable constants in a . 
Again, Q .Q and RP will each rise to the degree K + e+2- 1 in x ; but their sum 
is to be only of n~t dimensions in x. Hence we have to make {n+e+i- 1) _ {n >), 
; g I quantities (which are linear in respect to the given eoefficients in 
P and Q, as well as in respect to the new disposable constants in P, and Q,) a 
vanish, that is to say, there will be e+2.-l linear homogeneous equations to be 
satisfied by means of e + 2< disposable quantities ; the ratios of these latter are, theie- 
fore, determinate, so that we may write 
P, =\.(P,) 
Q, =\(Q,) 
R=x,(R,) 
and when (P,), («,), (R.) are taken prime to one another, it is obvious that (R) will 
be in all of e+2/ dimensions in the given coefficients, t. e. oi i in lespect o le 
coefficients of P, and of e+i in respect of those of Q: X, will correspond to what I 
have previously called the allotrioiis factor; being in fact foreign to the value of , 
as determined by means of the equation (4.), and arising only froiii the particiilai 
method employed to obtain it through the medium of the system (l.p it becomes a 
matter of some interest and importance to determine the values of this allotiious 
factor for different values of i*. 
. These ere iden.ioal with what I termed a«otients of succession in the London and Edinburg. PhilotopM 
Magazine (December, 1839) ; but by an easily explicable error of inadvertence, the quantities ' 
therein set out are not as they are therein stated to be. the quotients of succession or allotrious factor, them. 
selves, but the ratios of each such to the one preceding, if in the senes ; so that 
“ Qi” is A, 
Ao 
A, 
' Qo” is 
“ Q.q” is 
&c. . . . 
This error is corrected by my distinguished friend M. Sturm (Liouvilue’s 
theor^me d’Algebre de M. Syevester), who appears, however, to have over ooked that ^ ^ ' 
well acquainted with the existence and nature of these factors, and their essential character, of 
squares in the case contemplated in his memoir and my own. MM. Borcharut, Ferouem, ^ 
in quoting my formulae for M. Sturm’s auxiliary functions, have thus been led into the error o ai u in^ 
as completed by M. Sturm. 
