UNDER THE FORM OF A CONTINUED FRACTION. 
419 
terms, as evinced by the fact that there exist terms {ex.gr. adyi) having +1 for their 
numerical part. The same explicit method might be applied to show, that if the first 
divisor were e degrees instead of being only one degree in x lower than the first 
dividend, would be contained in every term of the second residue ; the difficulty, 
however, of the proof by this method augments with the value of e; but the same 
result springs as an immediate consequence from the method first given, which 
remains good mutatis mutandis for the general case, as may easily be verified by the 
reader. Applying now this result to the functions P and Q, supposed to be of the 
respective degrees n and n—e in x, and calling the coefficients of the leading terms 
in the successive simplified residues «!, a^, «3, &c., and the leading coefficient in Q a, 
and before denoting the successive allotrious factors by Xj, Xg, &c., it will readily be 
seen that 
1 1^^ 1 _ lo_ 
5je+l Xg.Xi— 2 X3.X2— 2 ^ 4 -^ 3 — 2i 
1 ^2 3 
e. 
* / 
and in general 
X.,: 
«i 
x.=- 
l 2 
«^«3 ’ 
X.„ 
* 3 ... 
1 „2 
\ _-,e+l 2 4 2m— 2 
CC r 
( 8 .) 
Art. (4.). Strictly speaking, we have not yet fully demonstrated that the complete 
allotrious factors are represented by the values above given for X, but only that these 
latter are contained as factors in the allotrious factors; we must further prove that 
there exist no other such factors. This may be shown as follows : it is obvious from 
the nature of the process that the complete residues will always remain of one dimen- 
sion in respect of the given coefficients, i. e. first of one dimension in the set a, h, c, 
&c., and of zero dimensions in a, |3, 7, &c. ; then conversely, of one dimension in 
a, |8, 7, &c., and of zero dimensions in a, h, c, &c., and so on, the residues being 
evidently required to conform in their dimensions to those of the first dividend and 
the first divisor alternately. These coefficients then are always of unit dimensions 
in respect to the given coefficients ; whereas it has been shown (art. 2.) that the 
simplified residues in respect to these coefficients are successively of the dimensions 
2 -{-e, 4-1-e, 6-f-e, &c. 
Let the complete residue corresponding to Xa^ be M.Xg^.aamj 
t. e. M.--2 — 
“1 “3 “5 
or say M.L; in passing from to the dimensions rise 2 units for all values of 
q except zero, and when q = 0 the dimensions increase per saltum from I to 2-1-e; 
hence the total dimensions of L in the joint coefficients will be 
((e+l)-(2e+2))- {m— l)4 + 4w + e= 1, 
