202 
ME. A. CAYLEY ON THE DOUBLE TANGENTS OE A PLANE CURVE. 
the first of which vanishes. On the right-hand side DA=0 identicaUy, and therefore 
D . A^H = A^DH, or we have 
(m-2) 
a. 
c, 
a' 
-\-{n—2) 
a. 
b. 
V i 
b, 
d. 
V 
b, 
c, 
d 
a!. 
d. 
d' 
d. 
v. 
b" 
:A==DH. 
+(%— 2)(w~3) 
b. 
d 1 
+ (72-2Xw-1) 
b, 
b. 
V 
c, 
d. 
b’ 1 
C, 
c , 
d 
b\ 
d. 
d' 
V, 
v. 
b" 
a. 
d. 
a 
+ (72 — 2 )(?^ — 2 ) 
a. 
c. 
V 
b. 
c , 
b 
b. 
d, 
d 
> 
1 
d. 
d', 
d' 
d. 
d. 
b" 
a. 
c. 
V 
+ (72 — 2 )( 72 — 3 ) 
a. 
b, 
d 
b. 
d. 
d 
b. 
c, 
d' 
d. 
d. 
W 
d, 
v. 
d' 
> = A^D^H; 
or collecting the different terms, 
(72— 2) (72— 3) 
a,b , d 
+ 2(72-2X 
a, c, b' 
1 
1 
a, d, d 
b , c , d' 
b , d, d 
b , e,b’ 
d, V, d' 
d, d, V' 
d, d', a" 
+ (72 — 2 )(?? — 1 );^. c. a 
iC. d. h’ 
\h'. c\ a 
17. A little consideration will show that in this equation we may write n-1 for «. 
and Hi for H. In fact, putting for a moment we ha^e coiiespoudiim 
to the equation 
this other equation. 
a , 
b. 
d 
= 
b. 
G, 
V 
d. 
b', 
a" 
la , 
lb. 
Id 
lb. 
Ic 
lb' 
Id, 
lb', 
la" 
= A^H, 
= A^H., 
where ultimately y,, z,) are to be replaced by {x, y, z). We may operate upon tins 
equation with D, D^ . . . as before, the only difference being that in the first instance 
la, lb. See. are as regards (x, y, z) of degrees lower by unit^^ than a, b. See., that is h- 1 
must be substituted throughout in the place of n', and when at the end of the process 
(+, y, ^/) are replaced by {x, y, z), then lb. See. become equal to «, b. See., from which 
the truth of the asserted proposition is manifest. 
