WHICH SATISFY GIVEN CONDITIONS. 
121 
91. In the case of the conic, (1), (2), &c. are the expressions denoted in the former 
part of this Memoir by (1 : :), (2 &c., the number of points being in each case such 
as to make in all five conditions ; calculating these functions by means of the formulae 
[a)=[_a\ See., the comparison of the resulting values with the values previously obtained 
will show a posteriori the limits within which the formulae are applicable ; where they 
cease to be applicable I find the difference, and annex it as a correction to the formula 
value : I have in some cases given what seems to be the proper theoretical form of this 
difference. We have 
(1 = 0 
= 2 
(2.*.) 
— a ; 
(3:) 
= — 4m— 3w+3a; 
(4.) 
= — 10m— 8w+6a ; 
(5) 
= -18m-15w+10a-[— 3m+a] (=-[/]); 
2(1, 1-0 
= (2 m+nj 1 
— 4m— n— 3a ; 
(1,2:) 
= (2 m-\-n)a 
+12m+12w— 14a ; 
(1,3-) 
= (2m+w)(— 4m— 3w+3a) 
+56m+49w— 39a ; 
(1, 4) 
= (2m~i~n)(— 10m— 8w+6a) 
+140m+122w— 84a 
— [(m— 3)(— 12m— 6w-|-6a)] (= — [(m— 3)(4/+2«)Jj ; 
2(2, 2 • ) = a 2 
-j-54m-j-48w — 40a ; 
(2, 3) = a(— 4m — 3%+ 3a) 
+144m+126w— 90a 
— [24m+6%4-(w— 12)a] (= — [Qr-\-{n— 3)*]) ; 
= (2 
+ 3(2 m-\-n){ — 4m— -n— 3a) 
— 32m— 58w>-|-78a 
— [4m(m — 1 ) (m — 2 )] ; 
6 ( 1 , 1 , 1 :) 
