504 MR. SYLVESTER ON A DEVELOPMENT OF THE METHOD OF ASSIGNING 
For greater brevity let [Q J [Q'J ['Q J be denoted respectively by 
then when the type Q, consists of a single element, 
^ '(yg=l 'iy;= 0 . 
It should be observed that the two equations ay ^~0 cannot exist simultaneously, 
for if represent 
&C., 
so that if 6;,=0 and «;= 0 , we have <y''= 0 , <= 0 , &c., and thus, finally, — 1 = 0 , which 
is absurd. 
Now, if we suppose to be types every elemenUn each of which is a 
linear function of the coefficients of x in these elements being positive in Qi, nega- 
tive in Q„ and so on alternately, and Q is the aggregate of it may easily be 
made out that each term in the development of cu in terms of ay^, ‘Ui, 'K i ‘>^■2, ^2, '1^2, ''^2 ; 
&c. will have the same sign when we give too; a value which is a superior limit, or an 
inferior limit to the roots of each of the cumulants a,, ... Ue, and consequently to 
those of the cumulants ‘u„‘co^,...'co,-, to/, ; the products affected 
with positive signs being all positive or negative in themselves, and those aftectei 
with negative signs being reversely all negative, or all positive. 
Thus, ex. gr. if Q=QiQ2 
and the sign of the leading coefficient in to^ will be the contrary of that in u.^, but 
and have both the same positive sign ; so again if Q=QiC22Q3, 
ly == (y 1 . &;2 . 6^3 . '<^2 • *>3 ”■ *^1 • <^2 • '"3 + *'1 • '<^2 • '<^3 J 
where the leading coefficients in a;2 and to2 have contrary signs, as have also those 111 
and«^ between ^2 and to^ have the same sign ; and of course the leading coefficients 
in m,, &^3, a;/, toe have all the same sign, they being all positive, and so in general. But 
the superior limit to the roots of any integral algebraical function of x substituted in 
place of a? causes the signs of the resulting values of the functions to coincide with 
the signs of the leading coefficients, so that in the example last above given, L a 
superffir limit to all the factors in the several products in the equation substituted 
for X will make -^ito^.tos to have all the same sign. Ihe 
like will be true of A the inferior limit ; for if Q2, contain respectively n,, n, 
elements, the values of the four products last above written, when x=- 00, will be 
to the values of the same when ^=+ co in the respective ratios of 
and so in general. Hence we deduce the theorem, that if the total type Q represent 
the ago-regatein apposition of the partial orders Q.Q2...^^e (the elements being under- 
stood to be linear functions of a:, which are subject to the law of alternation m t e 
signs of the coefficients of x in passing from one partial type to another), no supeiior 
