ivr-R. A. CAYLEY ON THE DOUBLE TANGENTS OF A PLANE CURVE. 
199 
a, b, 
b’ 
+ 
a, 
d 
b, c. 
d 
b. 
d. 
b’ 
d, b’, 
b” 
d. 
d. 
d’ 
values which, as I proceed to show, satisfy the identical equation 
Ia+m+IIW+IVe=0. 
11. We have in fact 
1= d{d¥~c'^ j^cd’-b’d’) 
{Vd -V’c-^-b'd-bd') 
-{•dJ{cd ~db'-\-hd’ — cd ), 
where the last line is 
bc¥-dVd!-, 
II = d {b'd — al'd + h'd — ¥c-\-a!d! — bd') 
-\-e [ca!' ~a!d ■\-b¥ —b'^ -\-ad' — a!d) 
-\-d'{a!d—b'c ■\-V c —bd -\-bd —ad!)^ 
where the last line is 
ddd!—ad !‘^ ; 
III= a{-b'dl^-cd'-\-d^ —¥'d->rb'd’-ea!') 
-\-b { bd’ —b’d-\-b”c—b’d-\- a!’d — a'd’) 
-\-a'{ cd —bd'-\-b’d—cd-\-a'e —b'd), 
where the last line is 
—bdd-\-eci ‘^ ; 
and 
IV = a{¥c—b'd-\-dd’—b'c') 
+5 (§'2 _iy<j^a!d -cdc) 
-\-d{bd — b'c -\-cb’ — dd), 
where the last line is 
bdd-d’\ 
And these values may be expressed as follows : — 
1 = «( 0 ) 
-f-J( ) 
+ (Z( ¥d^ d’c -b'd’ -dd- d'b' ) 
j^e{-b"c-d'b-\-b'd ■\-db' ), 
11 = ) 
+5(0 ) 
^d{-d'd-b"c-d'b-\-dd!^b'd^db'-\-d!d) 
+e{ d’c b”b + d’a — dd — b'b’ —dd ) , 
( 12 ) 
(13) 
(14) 
( 21 ) 
(23) 
(24) 
