198 
MB. A. CAYLET ON THE DOTJBLE TANGENTS OF A PLANE CTBYE. 
(i={x,y,zf ■ . (X',y', Z'), 
5'=(®,y,zf(X.Y,Z)(X',Y',Z'), 
c'=(®,y,2)(X,Y,Z«X',Y',Z'), 
d’= . . (X,Y,Z)‘(X’,Y',Z'), 
a"={x,y,zf . . (X',Y',Z7, 
b"=.{x,y,z) (X,Y,Z) (X',Y',Z% 
c''= . . (X,Y,Z/(X',Y',Z'f, 
where (X', Y', Z') are new arbitrary facients ; but, as before, (X, Y, Z) are taken to be 
current coordinates, and {x, y, z) the coordinates of the given point on the cuiTe : 
e = 0 is the equation of the curve ; 
(^=0, the equation of the first or cubic polar of the point {x^y, z); 
h =0, the equation of the last or line polar of the point {x, y, z), or what is the same 
thing (the paint being on the curve), the tangent of the cuiwe at this pomt; 
«=0, the condition which expresses that the point is on the curie. 
9. Imagine now an identical equation, 
Ia+IB+IIM+IVe=0; 
then, since «;=0, we have 
II5+IIM+IVe=0; 
and if in this equation we write 5=0, e=0, it becomes 111(5=0, that is, the pomts ot 
intersection of the curve e=0 and the tangent 5=0 lie on one or other of the cuives 
d=0, III=0. But the points in question do not lie on the curve (5=0, consequenth 
they lie on the curve III=0. 
10. To explain the law of formation of the multipliers I, II, HI, H , I form t ic 
matrix , 
o' ), 
d’ 
and then we have 
III=- 
( 
«, 5 
, c 
9 
(5; 
a' , 
5', 
5 , c 
, (-5, 
e ; 
5 ', 
d , 
a', h 
5 0 5 
d'-, 
a". 
5 ", 
d, c , 
V 
+ 
(5, 
5, 
d 
e 
, (5, 
d 
e , 
c. 
d' 
d 
5 ^ 9 
V' 
d\ 
5 ', 
d" 
1 (5, c, 
a! 
— 
( 5 , 
5, 
y . 
, d. 
y 
6 , 
c, 
d 
\ ^ 
5', (5, 
a!' 
d', 
5', 
b" \ 
a, 5 , 
d 
— 
a, 
c. 
y 
5 , c, 
d' 
5 , 
d, 
d 
a', 5 ', 
d' 
, a', 
d. 
5 " 
c" 
(5 , a, d 
e , 5 , d' 
d', a\ c" 
— ; a, (5, , 
5 , e , 5' 
a', (5', a!' i 
