ME. A. CATLET ON THE DOUBLE TANGENTS OF A PLANE CURVE. 205 
21. One other example will be sufficient to render manifest the law of the formation 
of the multipliers I, II, HI, IV . 
In the case of a sextic curve we have the matrix 
( a, b. 
c, 
d, e, 
a' , 
b', 
c' , 
d', 
b , c, 
d. 
b', 
c' , 
d', 
e' , 
a', b\ 
c', 
d', e\ 
a”, 
b". 
c'\ 
d!\ 
the identical equation is ^ ^ 
Ia+m+III/+IV^=0 ; 
and the expressions for the multipliers I, II, HI, IV are- 
1= 
/- 
e. 
V 
+ 
d , 
d 
+ 
c, 
d 
"h 
/. 
b , 
d 
•> 
/. 
d 
9^ 
e , 
d! 
d, 
e 
9’ 
c , 
f 
/. 
e'. 
W 
d'. 
d' 
d, 
d" 
b\ 
d’ 
H=- 
e, 
a! 
— 
d, 
V 
— 
f’ 
c. 
d 
— 
f. 
b, 
d! 
— 
a, 
d 
/’ 
b' 
9^ 
e, 
d 
S'^ 
d, 
d' 
9< 
c, 
d 
9^ 
b. 
f 
/'. 
a” 
d', 
W 
/. 
d, 
d” 
/. 
b\ 
d" 
a'. 
d' 
111= - 
a. 
d 
— 
a. 
c. 
d' 
— 
a , 
d, 
d 
a. 
d , 
b' 
— 
a , 
/’ 
a' 
b. 
c. 
f 
b. 
d, 
d 
b, 
e , 
d' 
i. 
d 
b. 
9^ 
V 
a\ 
b\ 
d' 
a!, 
d, 
d' 
a'. 
d', 
d' 
a!, 
d\ 
V' 
a'. 
/. 
a" 
IV= 
i 
b. 
d! 
+ 
a. 
c, 
d 
— 
a, 
d , 
b' 
— 
a , 
e. 
a! 
• 
i 
c, 
d 
b, 
d, 
d' 
b, 
e , 
d 
f. 
b' 
i 
b\ 
d" 
a',. 
d', 
d' 
1 
a', 
d', 
b" 
d, 
a!' 
22. We have in fact 
1 = « 0 
■\-b{—gd' 
-\-ff 
) 
(12) 
hJ-[-d'e^d"d-\-d'c 
_ df - dd -d'd’- dd -fb' 
) 
(13) 
-\-g{ — b"e— d'd — d"c 
j^de’-\-dd'-^d’d^db' 
) 
(14) 
a{ d'g 
-ff 
(21) 
+5 0 
• 
+/( _ df- We - d'd - d"e - 
■ d'b -fi a!f + b’d -\-dd’-\- d'd -fi db’ -\-f'a') 
(23) 
-\-g{ a"e-\-b"d-\-d'c-}-d"b 
— a'd — b'd' — dd — d'b' — da! 
) 
(24) 
: a{-b"f-d'e-d"d-e"c 
j^df^de'^-dld'+e'd^-fb' 
) 
(31) 
+ b ( a"f+ We -h c"d+d"c-\-e"b - a'f - b'd - dd' - d'd - db' -/< 
:f) 
(32) 
+/ 0 
• 
