476 MR. SYLVESTER ON THE QUOTIENTS RESULTING FROM THE EXPANSION 
C, means the coefficient of the highest power of ^ in R^, and Co the first coefficient 
in 
Aft ' (I ) If we takeg(*) of the same degree as/(*) and for greater simplicity make the 
first coefficients in f(x) and g(x), each of them unity, the successive simplified residues 
to 9 will be identical with the simplified residues to ^ (including amongst 
JX 
them the quantity gx—fx itself), and since 
{fx—g{x))^x— {fx—g{x))'gx= (g' xfx—f X .gx), 
the right-hand side of the equation above written, when the residues are made to 
refer to / and g, instead of referring to / and <p, are taken of the same degree in x, 
becomes equal to fxgx-fxg’x ; and if we now agree to consider / and g as homo- 
geneous functions each of the nth degree in x and 1, the equation becomes 
R? , R2 
Cl C1.C2 
Rs 
C2.C3 
-H 
r: 
Cn—\ • C„ 
^)}=l{^Ixf-^dlA){dxS) n{^dxS^dl‘s){dxf) 
1 
n 
dl'dx dx di 
where J indicates the Jacobian of the given functions / and g m respect to the 
variables x and 1, meaning thereby the so-called Functional Determinant of Jacobi 
to / and g in respect of x and 1, which equation also obviously must continue to 
hold good when we restore to the coefficients of x” in/ and g their general values. 
It may happen that for particular relations between the coefficients of / and g 
* This result may be obtained directly as follows : — 
Let fx, <px and the (m— 1) complete Sturmian residues be called po P\ P- 2 ---P’i’ ” complete quotients 
be called q^...qn, and Rt the allotrious factors to the residues p.^, p^, ... Pn be called ; then 
P0 = 9l-Pv — P2’ Pl=9<iP2—P3’ Pi = lzp3~P*' 
hence p^^pQ—po^i = p\h\ + ^p 2 ^Pi~Po^P^^ 
= P\hl + Pl^9'2 + i-Pi^Pi—Pi^Ps) 
= &c. 
=plhi+pm2+pl^93 + ■••-t-p'iMn ; 
but we have in general pi=/xi.Ri; 
hence 
Sq,=9^ .Hi^Sx 
C/j fxi 
Gj — 1 
and 1 
but it may be easily seen that 
1 
Ui-l'hi- 
cU 
hence 
plSqi= when i>l, and — when i — 1, 
* Cj-i.Cj C/, 
which proves the theorem in the text. 
