328 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Cayley, A. (1859, March). 
[On the double tangents of a plane curve. Philos. Trans. R. Soc. 
London, cxlix. pp. 193-212: or Collected Math. Papers, iv. pp. 
186-206.] 
The theorem on which an important part of this investigation rests is 
enunciated by its author as follows : If the 2n-l columns of the matrix 
a 0 
a x 
a 2 ... 
a n - 1 
; a 0 
«i • 
. . . a n _ 2 
a i 
a 2 
a 3 * • • 
a n 
; a'l 
a' 2 
. . . Cl n—1 
a 0 
a! i 
a 2 ... 
a 
; a 0 
a\ 
n 
, • • CL yi 2 
he represented by 
1 2 3 ... n ; (1) (2) ... (n- 1) 
respectively ; the determinant whose columns are those thus represented by 
r, s, (t) be denoted by {r, s, (t)} ; and the determinant aggregates 
{n,n-l, (2)} + {71,71-2, (3)} + ... + {ti , 2 , (n - 1)} , 
- {n , n - 1 , (1)} - {n,n- 2, (2)} - ... - {n,2,(7i-2)} - [n , 1 , (n - 1)} , 
-{1,2,(»-1)} - {1 ,3,(f»-2)} - ... - {l,7i-l, (2)} - {1,»,(1)}, 
{l,2,(*-2)} + {l,3,(n-l)} + . . . + {l,7i -1,(1)}, 
by I, II, III, IY ; then 
a 0 1 + cqll + a^III + aJN = 0. 
The mode of verification suggested consists in showing that there exist six 
quantities (12), (13), (14), (23), (24), (34), say, such that 
I = a 0 . 0 +a 1 (12) + a n _ 1 (13) + a n (14), 
II = -a 0 (12) + a 1 .0 + a n _ x (2Z) + a n (2i) , 
III = - a 0 (l 3) — aj(23) + a w _ t .O + a n (34), 
IY = - a 0 (14) - <^(2 4) - a n _ x { 34) + a n . 0 ; 
and then taking the sum of the requisite multiples. The six quantities in 
question are actually found for the cases where n = 3, 4, 6. In the last case, 
the matrix being 
a b c d e f a V c' d! e' 
b c d e f g b' c d' e' f 
a b' d d! e f a " b" c" d" e", 
their values are written by Cayley in the form 
(12) - -eg + ff, 
(13) = b"f+c"e + d"d + e"c - b'f - cV - d'd' - eV -f'V, 
(14) = — b" e — c' d — d" c + b'e + cd' + d!c + e'b', 
(23) = - a'f - b"e - c‘ 'd - d"c - e'b + af + b'e + c'd' + d’c' + e'b' +f’d, 
(24) = a"e + b"d + c'c + d"b - a'e — b'd' - dc' - d'b' - ea, 
(34) = - a" a + a' a ' . 
