OF THE DIFFEEENCES OF THE EOOTS OF A GIVEN EQUATION. 
53 
18. For the cubic equation {a, b, c, djv, 1)^=0, the equation for = y)'j is 0 = 
a‘ X 
9x 
27n X 
+ 1 
0 
+ 1 
c ■> 
+ 1 
19, For the quartic equation {a, b,c, d, e'X^, 1)^=0, the equation for /3, y)j 
is 0 = 
a® X 
96a- X 
A 
312\/ O X 
256 n X 
+ 1 
0 
< - ^ 
— 1 a?ce 
‘-1 a^d 
+ 1 
+ 3 a^<P 
+ 1 ab-e 
— 14 abed 
+ 9 ac^ 
-t- 8 ¥d 
- 6 b^d^ 
+ 3 abc 
-2 b^ 
ir- 
20. For the quintic equation («, b, c, d, e,fX% 1)®=0, the equation for /3, y, 
is 0= 
a’^X 
625 a®x 
12500 Ua^x 
A 
15625 □ X 
76125 n v' n 
A 
f » 
( ; 7^ 
( ^ 
r~ ^ 
+ 1 
0 
- 2 aV 
- 1 a^df 
— 3 a^e 
+ 1 
+ 24 a^def 
4- 4 
4-12 a^bd 
— 32 aV 
+ 3 a^bef 
4-16 
4- 2 
- 27 a^de 
— 50 aPe 
+ 264 a^bde^ 
— 36 aVe 
4-25 b* 
— 52 ePbeef 
- 96 a^’od/f 
+ 54 a^c(P 
- 2 aPf 
4-105 ab"ce 
I 
+ 64 cd(?df 
4- 352 aVeV 
— 180 abc^d 
— 936 o?cdPe 
-f- 80 uc^ 
4- 432 odd^^ 
— -50 b*e 
( 
1 
i 
1 
1 
4 - 28 ah^ef 
— 970 ab^ce‘' 
4 - 120 ald^dPe 
4- 264 aWedf 
4-2480 abc‘de 
-1440 abed^ 
- 192 ab(?f 
— 960 ac'^e 
4 - 640 ac^dp' 
- 160 bHf 
4 - 450 5V 
— 1400 b^ede 
4 - 800 ¥d^ 
4 - 120 5V/ 
4 - 600 lr<?e 
- 400 Vc^d? 
4-100 b^cd 
- 50 6 V 
. 
21. I remark, with respect to the equation in for the cubic, that it leads at once to 
the equation of differences. In fact we have 
a^^^+9(ac-by+^-27n=U,{l}~{a~(B)}, 
3IDCCCLXI. 1 
