WHICH SATISFY G-IVEN CONDITIONS. 
91 
all which is further explained by what follows; thus in the case (1)(1)(1)(1), 
=(l) Bli (l) BJa (l) Bl3 (l) m4 , the value of A is given as 1m{m 2 . w 3 m 4 (= 3m 1 m 2 m 3 m 4 ). Here A is 
the number of the point-pairs terminated one way in the intersection of any two m x , m 2 
of the four curves, and the other way in the intersection of the remaining two m 3 , m 4 of 
the four curves. But in the case (1, 1)(1)(1), =(1, l) m (l) mi (l) m . 2 , the value of A is given 
as=^m 1 m 2 +mmi.mm !! . Here A denotes the number of the point-pairs, which are either 
( lm x m 2 ) terminated one way at a node of m, and the other way at an intersection of 
m x , m 2 , or else (mm x . mm 2 ) terminated one way at an intersection of m, m x , and the 
other way at an intersection of m, m 2 : and so in other cases. 
42. This being so, we have 
(1)(1)(1)(1), =(l) OTl (lk(lk(lk 
A.-=%ni{m 2 ,m 3 m 4 (=3 m 1 m 2 m 3 TO 4 ), 1 
B =.%m x m 2 .m 3 .w 4 (=3Sm 1 m 3 m 3 w 4 ), 2 
C —'Zm x .m 2 .n 3 n i (= %m l m 2 m 3 n i ). 4 
A!=%n x n 2 .n 3 n i (= 3 %pz 2 % 3 w 4 ), 
B' = '%n l n 2 . n 3 . m 4 ( = , S%n l n 2 n 3 m i ), 
G — 2n,.n„. mjn, ( = 'Zn x n 2 m 3 m i ). 
(l,l)(l)(l),=(l,l) m (l) mi (l) W2 . 
A= hn x m 2 ~{-mm x .mm 2 ^ 
B= ln x m 2 + bi. 2 m x 
-\-mm x (n — 2 )m 2 -\-mm 2 (n — 2 )m x 
+ mm x n 2 (m — 1 ) + mm 2 n x (to — 1) 
-\-m x m 2 n(m— 2), 
C = Tm x m 2 
+nn x (m—2)m 2 -\-nn 2 (m—2)m l 
-\-n x n 2 .^m(m— 1), 
D= im x m 2 . 
(l,l)(l,l),=(l,l) m (l,l) m , 
A = + \mn x (mm x — 1 ), 
B= ln x (m x — 2) 2) 
-\-mm l (n—2)(m 1 —l)-j-mm 1 (n x —2)(m—l), 
C= r.\m x (m x — 1) -f 1 
+ nn x (m — 2 )(m, — 2), 
D=/.%m l (m 1 — 1) -\-i x .\m(m—l). 
1 A'= Tn y n 2 -\-nn x .nn 2x 
2 B'= Tm x n 2 -{-rm 2 n x 
-\-nn x (m—2)n 2 -\-nn 2 (m—2)n x 
-\-nn x m 2 {n—V) -\-nn 2 m x (n—l) 
-\-n x n 2 m(n— 2), 
4 C' = 'hlfh x Yl 2 
+mm l (n-2)n 2 -\-mm 2 (n—2) 
+m x m 2 .^n(n—l), * 
3 D'= «WjW 2 . 
1 A'= tt x -\-\nn x (nn x —} 
2 B'== rm x (n— 2) +r x m{n— 2) 
-\-nn x {m—2)(n x —V)-\- nn x (m x — 2) 1 
4 C'= ^.^n x (n x — 2) 1^ 
-f-mTOi(w- 2X^—2), 
3 D'= *.-iw 1 (» I -l) +^.^(w-l 
MDCCCLXVIII. 
P 
