ME. A. CAYLEY ON THE DOUBLE TANGENTS OF A PLANE CUEYE. 207 
and we have to form the identical equation 
25. If, for shortness, the columns of the last-mentioned matrix are represented by 
1, 2, 3...7^,(l), (2)..(^-l), 
and the determinants formed with these columns respectively by a corresponding nota- 
tion {1, 2, (1)}, {1, 2, (2)}, &c., then the expressions for the multipliers I, II, HI, IV 
are as follows, viz. 
1= {n^n — I, (2)} -l-{w, w — 2, (3)} . . . 2, (%“!)} , 
n=— {w, n—\, w— 2, (2)} . . . —{n, 2, {n-2)}~{n, 1, 
III=-{1, 2, (w-l)}-{l, 3, (w-2)} ... -{1, ^^-l, (2)}-{l, (1) }, 
IV= {1, 2, (w-2)} + {l, 3, (w-1)} ... +{1, (1)} ; 
the truth of the identical equation being shown, as in the foregoing special cases, by the 
transformation of the multipliers into the form 
I=«o 0 H"<*i(12)-j-^n-i(13)-l-®n(14), 
II=<21)+a, 0 -f«„_,(23)+<24), 
IIl = «o(31)+cq(32)+«„_, 0 +<34), 
IV=«„(41)-t-«,(42)+«„_i(43)+«„ 0 , 
where (12)=— (21) &c. : the required expressions may be written down without diffi- 
culty. 
26. Proceeding then to reduce the equation 
III=-{l,2,(7i-l)}-{l, 3, (w-2)}...-{l,w-l,(2)}-{l,w,(l)}. 
we have the equation 
{1,2, (1)} = AHI, 
which is to be successively operated on with D. The degrees (less unity) of the 
columns 
1, 2, ..w— 1, w, (1), (2), ..(w— 1), 
are 
w— 1, w— 2, ..1, 0, w— 2, w— 3, .. 0; 
and the rule is to operate on each column of the determinant, multiplying by the degree 
less unity, and increasing the symbolical number by unity. Thus 
D{1, 2, = 2, (1)}H-(7z- 2){1, 3, (1)} +(w-2){l, 2, (2)} 
= (^-2){1, 3, (1)} +(^^~.2){1, 2, (2)}, 
since (2, 2, (1)} vanishes identically. The following Table shows the mode of effecting 
the operations : — 
