THE THEOET OF EQUATIONS OF THE FIFTH OEDEE, 
271 
Coeff. 
= _ 
-j- 4(2^a/3y2a 
-16aV2a/3 
+ 64aV. 
= —{lQace—Sad^—8b^e-\-ibcd~cY 
-\-iace{16ace—8ad^—8b'^e-\-4:bcd—c^) 
— 16«V(— 16«e+45(Z~c^) 
+64«V. 
21. Effecting the developments, these are 
-32 
ae 
+ 240 
9 9 
a-^e 
+ 320 
+ 8 
bd 
— 
112 
abde 
— 
64 
a‘bde^ 
— 3 
c’ 
— 
8 
ac‘e 
— 
176 
+ 
16 
acd^ 
O- 
J 
224 
ared'^e 
+ 
16 
b“ce 
— 
64 
a^d* 
+ 
16 
¥d^ 
+ 224 
ab^ce^ 
— 
16 
b(Pd 
— 
128 
ab^d“e 
+ 
3 
— 
112 
abc"de 
64 
abc(P 
+ 
28 
ac^e 
— 
16 
ac^(P 
— 
64 
b^e^ 
+ 
64 
b^ede 
— 
16 
b^c^e 
— 
16 
bW 
+ 
8 
bc^d 
— 
1 
the fii’st of which is in fact complete ; the others being completed, the equation in (p is 
found to be 
22. For the denumerate form {a, b, c, d, e,f^v, 1)^=0, the equation in <p is 
X 
X 
— soov" Sa^v^n X 
A 
A 
A 
A 

— 32 ae 
0 
O 
O 
1 
r - - 
+ l 
+ 4000 aW"' 
+ 8 bd 
+ 240 o'^e’ 
— 
1600 (Pdef 
— 3 c' 
+ 240 abc/ 
. 
+ 
320 a^e^ 
— 112 abde 
— 
1600 aWf 
— 8 ac^e 
— 
640 edbeef 
+ l6 aed^ 
+ 
640 cdbdrf 
- 64 by 
— 
64 (Pbde^ 
+ 1 6 b'ce 
— 
80 arc^df 
+ 16 b^^cP 
— 
176 «W 
- 16 bc^d 
+ 
224 a"ccPe 
+ 3 
— 
64 a^d^ 
+ 
384 ai'V 
— 
192 aVedf 
+ 
224 aVee^ 
— 
128 aVdh 
+ 
48 abey 
— 
112 aberde 
+ 
64 abed^ 
+ 
28 ad'e 
— 
16 add^ 
64 6V 
+ 
64 b^ede 
— 
16 dc^e 
16 6'cV* 
8 bc‘^d 
— 
1 c« 
WT=o, 
where □, =.a*f*-\- See., denotes the discriminant for the denumerate form. 
