UNDER THE FORM OF A CONTINUED FRACTION. 
4-27 
If we equate the right-hand members of (e-{-2/) of the above equations to zero, 
and then eliminate dialytically the several powers of x from to (both 
inclusive), the result of this process will evidently be of (e-j-<) dimensions in respect 
of the coefficients in V, and of / dimensions in respect of the coefficients in U ; and of 
the degree in x it will also be of the form 
. . .La7®~')U-l- (F-j- G.r-f- . . . , 
and by virtue of art. (2.) must consequently be the /th simplified residue to the 
system U, V. 
Art. (10.). The most general view of the subject of expansion by the method of 
continued division, consists in treating the process as having reference solely to the 
two systems of coefficients in U and V, which themselves are to be regarded in the 
light of generating functions. To carry out this conception, we ought to write 
17=^0 -}-n!3?/^“f“&c. od inf. 
V = 5o + ^i-,y+^2.3/"+%"+&c. ad inf., 
and might then suppose the process of successive division applied to U and V, so as 
to obtain the successive equations 
U-M,V+R,=0 
V 1^2=0 
Rl M3R2+R3=0 
&c. &c.. 
Ml, Mj, M3, &c. being each severally of any degree whatever in y, and in general 
the degree of 3/ in M, being any given arbitrary function <p (/) of <. The values of the 
coefficients of the residues Ri, Rg, R3 . . or of these forms simplified by the rejection 
of detachable factors, becomes then the distinct object of the inquiry, and will, of 
course, depend only upon the coefficients in P and Q and the nature of the arbitrary 
continuous or discontinuous function ^(/), which regulates the number of steps 
through which each successive process of division is to be pursued. Following out 
this idea in a particular case, if we again reduce to our two initial functions the forms 
previously employed, and write 
U=ao-‘^”+ +&C. 
and if, instead of making, according to the more usual course of proceeding, the 
divisions proceed first through one step and ever after through two steps at a time 
which is tantamount to making ^1 = 1 (p{\-\-a) = ' 2 , we push each division through one 
step only at a time, and no more (so that in fact <p{i) is always 1), we shall have 
U — m^. V -|-Ri=0 
V — m^x. Ri- 1-R2=0 
Rl — m 3 . R2-TR3=0 
R 2 — m4.^.R3-l-R4=0 
&c. &c., 
