CURVES WHICH SATISFY GIVEN CONDITIONS. 
161 
— 348m 2 — 696m%— 348% 2 -f- 2640m -f-2640%-|-a( fm 3 -f- 2 m 2 % + 2m% 2 +f% 3 <d 
f-m 2 +— 3 ^-mw + & xf% 2 — 3159m— 3159% + «( — fm 3 — ^ntfn — ^mn 2 — f % 3 w 
— i 2 % 2 - 14m%— -^f-% 2 — J ^m+ ~f%+“( ^m 3 + m 2 %+ m% 2 +f% 3 <s> 
_ 3|^ m 2_ 3|% m - iL tt% 2 + ^f%+«( fm 2 %+ 3m% 2 +f% 3 <e 
— 4|-|%i 2 — ^f% 2 + - 7 -f%+a( fm 3 + 3m 2 %4- fm% 2 <*> 
+ -^m%+ -^% 2 — 150%+a( (6) 
— 15m 2 — 52 m%— 15% 2 -}- 1920) +a 2 (— 3m— 3%-f56) 
+- L |- L m 2 +115m%+- L f% 2 — ia /%+2430) + a 2 ( ^m+^%— 75) 
— f m 2 — m%+ f% 2 — fm— ~ 3 -%— 34)-j-a 2 (— -fm— -f%+13) 
— 4m 2 — hnn— - 4 % 2 + • L pm+ 1 ~ i 2 ^ — 238)-fa 2 ( — f%+ 3) 
— -^-m 2 — - 2 -m%— 4% 2 + i ff 1 m+ -f%-— 238)+a 2 ( — fm + 3) 
— 3m%— 3% 2 + 29% ). 
Verification is 
(-i-t-¥-3-i)8+(4+¥+¥+9)4+G¥-i4--H*)2+(--¥+*F) 
+ 3((|+l + l+i)8+(-f-l+f)4 + (-|-¥)2-34) + 9((-|-|)4+13.2) = 0. 
126. It will be observed that in the eighth and following equations, viz. those 
wherein the expression of the Supplement contains the symbol (1), I have included along 
with the Supplement within the { }, the terms — (m — f {2( • 4) — (/4)} &c., viz. these 
are — (m— f) into number of point-pairs (4), &c. : this is for convenience only; it sim- 
plifies the calculation, both from the symmetrical form under which the remaining terms 
present themselves in the several equations, and because the expressions of the terms 
in question, (these terms being mere multiples of a number of point-pairs) are by 
Zeuthen’s theory known in terms of the Capitals. It is to be noticed that for any 
equation, to find the system to which the Capitals belong, we diminish by unity the 
barred number and then remove the bar ; thus for the seventh equation, where we have 
Supp. (2, 1, 1, 1), the Capitals belong to the system (1, 1, 1, 1). 
