502 MR. SYLVESTER ON A D VELOPxMENT OF THE MEIHOD OF ASSIGNING 
auxiliary constants, this coincidence may be brought about, so long as theie i. a 
sino'le real root in fx. ^ rr> • 
h is rather important to demonstrate this universal possibility o e ecting a coin 
cidence of the limits to the roots with the extreme roots themselv-es, because i i 
most striking feature which distinguishes the method of limitation here deve p 
from all others previously brought to light. 
Art. («.). Before entering upon this demonstration I may make ihe passing remaik, 
that every method of root-limitation is implicitly a method of root-approxima ion 
For instance, let e be any given quantity between which and it is known that 
a root of fa lies. Then if we write ir=e-l-i and form the equation -0, 
ami find L a superior limit to y, it is clear that e-ki will lie between e and the root 
of fa say E, next superior to e. Again, making a;=e-kL+“o and finding a -up 
limit L' to y’, we shall have e-k^+p st'H nearer to E than c-kj^ "a® . 
proceed advancing nearer and nearer, and always from the same side towards E at 
each step, and finally obtain E under the form e+x+L'+I7'+ 
manner calling the root next below e, we may find 
Art. (/.). In establishing the theorem of coincidence above adveited to, 
ing notation will be found very advantageous. Let Q denote a 1 ype o anj 
of Elements, as ?„ and let fk denote this same 
and 'Q the same type when the first element ,s cut off, ai d Q the same ‘ 
both extremes are cut off so that the apocopated type Q will mean [ 5 , 
apocopated type's will mean [5, ,....9.], and the doubly apocopated type Q 
'"Tniw 'a typ^Q be made up of the types Q. Q,...aiput in apposition, 
in general [Q] to denote the cumulant corresponding to the type , t leie n i 
simple law* connecting [Q] with 
['Q'j ['□;]... 
This law will be seen to be obviously dediicible by successive steps of expansion 
The cumulant corresponding to any portion or fragment of a type may be s^aid^to be a i r i 
the entire type, and a type whose elements are constituted out of the e emen above may then be 
juxtaposition may be said to be the aggregate of these types; the au gi complete and partial 
said to have for its object the expansion of the complete cumulant to any type n 
cumulants to the types of which the given type is the aggregate. 
