G08 
PROFESSOR CAYLEY’S TERTH MEMOIR OR QUARTICS. 
R.G.F. of 
quin tic = 
Deg-orders. 
1 . 
1— 6 5 
0.0 — 10.10 
+d . 
1 — ag* 
3.3 — 12. 8 
+ e . 
1 lr 
3.5— 7. 9 
+/ • 
1-6 
3.9— 5.11 
-\-h . 
-t o 
1— 1 W 
4.4—13. 9 
+i • 
1-6 *g 
4.6 — 12.10 
+./ • 
1—ag 3 
5.1 — 14. 6 
. 
1 — 6 2 
5.3— 9. 7 
+ 1 ■ 
l-bg 
5.7 — 11. 9 
+m . 
1—cuf 
6.2 — 15. 7 
+ n . 
1-1 Ag 
6.4—14. 8 
+ 0 
1 — 6 3 
7.1 — 13. 7 
+P • 
1—6^ 
7.5 — 15. 9 
+ r . 
l—lfig 
8.2 — 16. 6 
+ dg . 
l-ag* 
8.4—17. 9 
1 — abg 
9.3 — 16.10 
+4? • 
i —<qr 
9.5 — 18.10 
1 — ag 2 
10.2—19. 7 
1—6 >g 
10.4 — 18. 8 
~\~t 
1 — 6 3 
11.1—17. 7 
+jm . 
9 , 
1 a 9 
11.3 — 20. 8 
+jo • 
1 — bg 
12.2 — 18. 4 
+ y . 
1 — 6 5 
13.1 — 23.11 
+,/' s ' • 
1 — bg 
14.4 — 20. 6 
J rJ t • 
1 ~9 
16.2 — 20. 2 
+ M • 
1 —a 
18.0—19. 5 
I —a . 1—6 . 1— c . 1 —g . 1 — q . 1 — u 
where observe that each negative term of the numerator is equal to a positive term 
into a power or product of terms a, 6, g, contained in the denominator : this is the 
condition above referred to. The expansion thus consists only of terms each with the 
coefficient -f 1 ; for instance, a part of the function is 
■s (1 — abg) s 1 — abg 
l—o.l — b . 1 — c . 1 —g . 1 — q . 1 —u 1 — e . 1— <? . 1 — u ’ 1 —a . 1—6 . 1 —g 
where the first factor is the entire series of terms sc s q%^, and the second factor is the 
series of terms a a b l3 g y omitting only those terms which are divisible by abg : and in the 
product of the two factors the terms are all distinct, so that the coefficients are still 
each = 1 . The same thing is true for every other pair of numerator terms : and (since 
