FOE AN EQUATION OF ANY OEDEE. 
97 
and consequently 
= L,= -iv'L„ L.=-iv'L„ &c., 
gi\’ing 
Lj= — ^{( — 6+24=)l ^abcf' — } = — 1 8aJc+ 46®, 
L,= - -54«^c+(36-36 = )0 «6^} == +27a^ 
L3 = 0, &c., 
and consequently for the last coefficient the value above written down ; it will be pre- 
sently seen how in more complicated cases the calculations should be arranged. 
Again, multiplying together the equation of differences and the equation for the 
squares of the roots of the cubic equation, we obtain an equation which it is not neces- 
sary to wTite down, as it can be at once formed by putting c=0 in the equation of 
differences for the quartic equation. And from the equation so obtained, by the— 
adjunction of the terms in e, we find the equation of differences for the quartic equa- 
tion, -viz. each coefficient is of the form Lo+Lje-l-LoC^-l- &c., where L,, is known, and 
such coefficient is reduced to zero by the operator 
4ff^j-|- 369^-1- ; 
or putting for shortness V'=4aBj-}-36Be-}-2c^^, the operator We have there- 
fore 
h, = -lv'L„ L,= -4v'L„ L,= -iv'L„ &c. 
It is to be observed that the last coefficient of the equation of differences is the discri- 
minant, and that the above method of calculating the coefficients of the equation of 
differences, as applied to the last coefficient, is nothing else than the method of calcu- 
lating the discriminant given in my Fourth Memoir on Qualities. 
The multiplication of the equation of differences, and the equation for the squares of 
the roots of the quartic equation, gives, in like manner, the equation of differences for 
the quintic equation, except as to the teians involving f ; and these are obtained as 
above, \iz. each coefficient is of the form Lo+L^-j-L^y^d- &c., where Lo is known; 
such coefficient is reduced to zero by the operator 
oadj-j- A:l)bg-\- ; 
or putting for shortness V'=5«B4 + 46B,+ 3cc)<^+2fZB„ the operator We have, 
therefore, 
L, = -iv'L„ L.= -iv'L„ L.= -iv'L„&c., 
which give L,, Lj, &c. by means of L,,. For the calculation of V'L (where L is 
any one of the coefficients L^, Lj, &c.), it is proper to separate the terms involving 
