WHICH SATISFY GIVEN CONDITIONS. 
97 
(2) 
(•’•)= 
( :/ )=2 «, 
(•//)=2«, 
(///)= «; 
(2,1) 
( : )=12m+12w-|-(2m-f- w— 14)a, 
( •/ )=24m+24#+(2m+2w— 24 )a, 
( // )=12m+12n + ( m+2w— 14)a; 
(2,1,1) 
( - )=24m 2 + 36wm+12# 2 -168m-168w+a( m 2 +2mM-^ 2 -25m-f>%+138)- 
( / )=12m 2 +36mw + 24w 2 -168m-168w+a(fm 2 -f-2mrc+ n 2 -^m— 25ra+138)— 
(2,2) 
( • )=27m+24rc-20a+±a 2 , 
( / ) = 24m + 27w-20a+ia 2 ; 
(3) 
( : )= — 4m— 3w-j-3a, 
( ■/ )= — 8m— 8w-f6a, 
( // )= — 3m— 4w+3a ; 
(3, 1) 
( • )= — 8m 2 — 12mw— 3w 2 +56m+53w-f-a(6m-f 3w— 39), 
( / )= — 3m 2 — 12mw— 8% 2 -f53m-f-56ft+a(3m+6w— 39); 
( 1 ) 
( • )= — 10m— 8w+6a, 
( / )=- 8m — 10% + 6a. 
50. By means of the foregoing formula; I obtain, as will presently be shown, the fol- 
lowing formulae for the number of the conics which satisfy five conditions, viz. : — 
(5)= — 15m— 15%+9« ; 
(4. 1) = — 8m 2 — 20mw— 8w 2 -f 104m+104w-f a(6m+6«— 66); 
(3.2) = 120m+120n+«(-4m-4w-78) + 3 a 2 ; 
(3, 1, 1 )= — fm 3 — 10m 2 w— 10»m 2 — fw 3 4- i ipm 2 +116mrc-f i f% 2 — 434m— 434n ; 
+ a(fm 2 + Qmn + f n 2 — — -^n + 2 9 1 ) — fa 2 ; 
(2, 2, l)=24m 2 +54mw+24«, 2 — 468m— 468w 
+a(— 8m— 8^+327) +a 2 (fm-+-fw— 12) ; 
