96 
ME. A. CATLET ON THE EQHATION OF DIFFEEENCES 
For the quartic equation, 
Equation of differences is, 0 = 
A 
a* X 
c? X 
A 
A 
A 
( V 
( ^ ^ 
( 
f ^ 
( ^ 
/ 1 
f M 
+ 1 
+ 8 «c 
-f 8 a^e 
+ l6 a'^ce 
-112 aV 
— 192 a'c^ 
+ 256 ff’e" 1 
-3 b- 
— 2 cPhrl 
+ 26 
+ 56 a^bde 
+ 216 o?d~e 
— 192 a'bdd I 
+ 22 aV 
— 6 a^b'^e 
+ 24 aVe 
+ 72 a^b-^ 
— 128 a-dd 1 
— 16 ab~c 
-30 a^ed 
+ 48 a^c>F 
— 120 arbcde 
+ 144 (TCCFe 
+ 3 5" 
+ 28 aV 
— 32 a^b’^ce 
— 54 a'bd^ 
- 27 aV" 
+ 8 a-Pd 
- 25 d-b-d^ 
+ 32 a-c^e 
+ 144 ah'cd 
— 24 arb~c" 
— 54 d^b<?d 
+ 18 a"c~d- 
— 6 ab~d-e\ 
+ 8 ab^c 
+ 17 a-c" 
+ 18 ab^de 
— 80 abdde\ 
- 1 5« 
+ 6 ah^e 
— 6 ab-<?e 
+ 18 abcd^ 1 
+ 38 ab^cd 
+ 42 ab'cdr 
+ 16 ade I 
- 12 ab-c^ 
- 26 abc^d 
— 4 add- \ 
- 6 b^d 
+ 4 ad 
— 27 b*d ' 
+ 2 5"e= 
- 9 b*dr 
+ 18 dcde I 
+ 6 b^dd 
- 4 ' 
- 1 b-d 
— 4 b'c^e 1 
+ 1 b'dd'- 
Equation for squares of the roots is. 
( 
or X 
A 
A 
A 
A 
■ 
A 
( 1 
+ 1 
r 1 
+ 2 ac 
-1 b^ 
! 
+ 2 ae 
— 2bd 
+ 1 c= 
( 
+ 2 ce 
-1 d^ 
f > 
+ 1 
ly. 
The multiplication of the equation of differences and the equation for the squares of 
the roots of the quadric equation, gives the equation, 0 = 
where all the coefficients except the last are reduced to zero by the operator 
and they are consequently (without any alteration) coefficients of the equation of differ- 
ences of the cubic equation : the last coefficient is not reduced to zero by the operator, 
and requires therefore to be completed by the adjunction of the terms in d (the series, 
here and in every other case, is of course a finite one, the number of terms might easily be 
calculated Let the value be &c., we have Lo= + 4ac®— : 
and putting for shortness V'=3aBA-l-25Bp, the operator which reduces this to zero is 
we ought therefore to have 
X 
A 
A 
^ 
< ^ 
+ 1 
r 
+ 6 «c 
-2 5" 
+ 9 a-d 
— 6 ab'^c 
+ 1 d 
r 
1 Q ^ 
+ 1 
0= V'Lo+ d 
+ cL\ 
V'L,-f d'^l V'L,+ .. 
2cL2 I 3cL3 
