FOE EQUATION OF ANY OEDEE. 
105 
quartic equation («!, c, d, e%y, 1)^=0, viz. these coefficients are 
c* X &c. 
- 1 - 
—2a^hd 
4-22aV 
— IQab^c 
+ 35^ 
and by writing b, c, d, e, f in the place of «, 5, c, d, e respectively, and multiplying by 
b^, we have the above-mentioned series, 
-j-lJ®, -^-Wd, &c. 
-3JV. 
a” X a*x 
-J- 1 -f- 8<zc 
-W 
It is easy to see, d priori, in the case of an equation of any order, that this property 
holds good. 
Passing now to the standard forms, — 
For the quadric {a, b, cjjv, 1)^=0, the equation of differences is, 0= 
1 ^, 1 ). 
For the cubic equation {a, b, c, d'^v, 1)®=0, the equation of differences is, 0 = 
1^. 
X 
18 X 
A 
81 X 
A 
27 X 
r 
+ 1 
+ 1 
f „ o 
-t - 1 a‘c^ 
— 2 ab^c 
+ 1 b* 
' +1 
— 6 abed 
+ 4 ac^ 
+ 4 bH 
-3 
C? X 
A 
4 X 

( \ 
+ 1 
1 + 

For the quartic equation {a, b, c, d, e^y, 1)^=0, the equation of differences is, 0 = 
a*x 
48a^ X 
8a- X 
A 
32 X 
A 
16 X 
A 
1152X 
A 
256 X 
A 
\i \ 
+ 1 
! 
f ^ 
-f 1 ae 
-1 
f I 
+ 1 
— 4 (^bd 
-t- 99 aV 
-192 ah\ 
+ 96 b^ 
t 
Z a*ce 
-i- 13 a^d^ 
— 3 (^b^e 
— 90 (j?bcd 
+ 189 ffV 
+ 64 e?Wd 
— 432 aW 
+ 384 afre 
-128 U- 
< 
— 7 aV 
+ 36 a^bde 
+ 54 a^cre 
+ 288 a^cd‘ 
— 192 a^b^ce 
— 400 aW-d^ 
— 1944 edbe^d 
+ 1377 
+ 96 ab'^e 
+ 3648 ab^cd 
-2592 a6V 
-1536 bH 
+ 1152 
( ^ 
— 1 a^ce‘ 
+ 3 d^d^e 
+ 1 a^b^e^ 
— 10 d^bede 
-12 d^bd^ 
+ 6 d^c^e 
+ 9 d^c^d"^ 
+ 4 ab^de 
— 3 ab'^c^e 
+ 56 aWed"^ 
-78 abdd 
+ 27 «c® 
— 32 ^d"^ 
+ 48 
-18 6V 
r 'i 
+ 1 
- 12 d^bdd 
— 18 a‘<rd 
+ 54 a”cd‘e 
- 27 a\V 
+ 54 ab^ce‘‘ 
— 6 ab‘^d'^e 
— 180 abc‘de 
+ 108 abed^ 
+ 81 uc^e 
- 54 
- 27 
+ 108 Wede 
— 64 
— 54 b~de 
+ 36 b^cH'^ 
