WHICH SATISFY GIVEN CONDITIONS. 
89 
we have in the first case 
(2X .•.)= -h'+¥ 
— — fo / 
+2 fjj'— 1 ) 
=p!+W-V+W-J),=t*', 
and similarly in the other three cases, 
(2X •//)=>', (2X •//}=/, (2X///)=<r f . 
38. Let (jM/, r, g, ff) be the characteristics of 2Z, ((««— fv+ff — <r=0), and (^', i/, g>', cr') 
the characteristics of 2X, ((jJ — -fV+fg' — ^ = 0). Then in the formula for (2X, 3Z), 
writing successively for 3X 
(2X-), characteristics Qm, v, g), 
and 
(2X/), „ (,, e ,*), 
we obtain expressions for the characteristics (2X, 2Z •) and (2X, 2Z/) of (2X, 2Z), viz. 
eliminating from the formulae, first the (a. </) and secondly the (p, ;jJ), each of these may 
be expressed in two different forms as follows : — 
(2X, 2Z • ) 
+iW'+^ f s) 
-hi 
+U v i+ v '& 
(2X, 2Z/) 
-iff' 
+W+ l/ ?) 
is’®' 
+K w '+* /ff ) 
~U?i+h\ 
the two expressions of the same quantity being of course equivalent in virtue of the 
relations between (^, v, g, a) and (p/, v', g', <r') respectively. 
The characteristics of (X, Z), (X, 2Z), (X, 3Z) are at once deducible from the before- 
mentioned expression ap,- \-(3v of (X, 4Z). 
39. Zeuthen’s investigations are based upon the before-mentioned theorem, that in a 
system of conics (4X), characteristics (p», v ), there are 2p — v point-pairs and 2v — p, line- 
pairs. If in the given system the number of point-pairs is=X and the number of line- 
pairs is =w, then, conversely, the characteristics of the system are 
^=J(2X+ ts-), v=|(\+2®-). 
