ON THE QUOTIENTS APPERTAINING TO THE EXPANSION OF 
fa 
Alt. (c.) Let r....ri be used in general to denote the determinant 
+ l •yri+2 • • • 
469 
->v. 
’■i+I 
A + 2 • • • 
then the simplified ith Sturmian residue R.- may be expressed under the form 
Di, 2, 3 . . . *■ a:”-'-* - Da, 3, . . . i+i ^"-‘'-"+ 03 , 4 . . . . . 4:D„_,, 
which is easily identifiable with the known expression for such residue. 
Novv obviously the necessary and sufficient conditions in order that the n roots 
may consist of only repetitions of i distinct roots is, that R, shall be identically zero, 
that is to say, we must have 
^1,2, Da 3 0 D — 0 
But the reasoning of the preceding article shows that although these equations are 
necessary and sufficient, they are but a selected system of equations of an infinite 
number of similar equations which subsist* and that, in fact, whatever be the value 
of {n), we may take r, r^...ri perfectly arbitrary and as great as we please, and the 
equation 
r>r r r =0 
must exist by virtue of the existence of the n—i equations last above written. 
Art, (</,) I now return to the question of expressing the successive quotients of 
as functions of the differences of the roots and factors ; that they must be capable of 
being so expressed is an obvious consequence of the fact, that the numerators and 
denominators of the convergents have been put under that form, since if 
N.--2 
Di_.; Di_? Di 
are any three consecutive convergents of the continued fraction 
1 J_ 
Di~ Q2 — Qj’ 
we must have 
d,_2.n,-n,_2.d,=q,. 
It would not, however, be easy to perform the multiplications indicated in the above 
equation, so as to obtain Q, under its reduced form as a linear function of I pro- 
ceed theiefore to find Q^- constructively in the following manner. 
Let R;_2, R._„ R. be three consecutive residues, /'.r counting as the residue in the 
zero place, then Q,=%^ and is of the form 
q I q! 
But qucere whether any other sufficient system can be found of equations so few in number as this system. 
