470 MR. SYLVESTER ON THE QUOTIENTS RESULTING FROM THE EXPANSIO*’ 
Now in general if we call the n roots of /r, where the coefficient of x" is supposed 
tollity! K...K, and if we use Z.to denote with the convention 
that Zi=w, Zo=l,wehave, employing (i) to denote I V+i 
R. = 
Zf_j.Zt2.-Z«+i‘ 
”z?_,.zt3...z« 
^ _^Zt -3-Zl.5--Z 
Z?_2.Z|_4.--Z(,)+1 
The part of R^-i within the sign of summation is 
2(A9i+i-!-^«i+2+-”+^0^(^«i he,’--h)x^ +&C., 
z,x”-^-z;x»-^-+&c., 
and the part of R,_2 within the sign of summation is 
z,_iX™-*+‘--z;-_iX”-'+&c., 
, ^2 Z,:-,x>^-^+^-Z;-.x^'”-- ^__7. ^ Z,.x4-fz.-,,.z:.-z,z;_0+ an algebraic fraction. 
[ Zi_ 2 • Zi-4---Z(,-)+l ' 
Hence 
Qi r7'2 ’ 1^2 r?2 r7'2 
^)- 
■ zflTzt 3 • • • Zw+ 1 ' lz|_ 1 . ZLs • • • Z( 
X {Zi_i.ZiX+(Zi_i Zj Zj Zi_i) } 
__zf-i Zt-3.Zt-5---Zffl rp 
“ zf ’zU^zU^TT^/ 
Ti denoting Zi_,.ZjX+(Zi_iZ;—Zi.Zi_i). 
Art (e ) If the process of obtaining the successive quotients and residues be con- 
sidtr^ it will easHy be seen that each step in the process imports two new coeffi- 
cients into the quotients, the first quotient containing no literal quotient in t ,e 
nuiltiplving :r and containing the first literal coefficient in the other part ^ ^ 
qnotien't containing two literal coefficients in the one part ^ 
in general the ith quotient containing 2i-2 of the P‘ , 
of them in the other. Hence T, being made equal to L,.a-1-M„ L, conta 
Mi contains 2i— 1 of the literal coefficients of /x. 
Moreover, we have Pi_ 2 -mPi 
Zi of the form T^-^7;7"‘ > 
Pi_i = 2^(Apj }l0„...he^^ei+i 
, : it »ill be remembered is the symbol ot tbe operation of ..bing the product of the squares of the differ- 
ences of the quantities which it governs. 
