104 
PROFESSOR CAYLEY ON THE CURVES 
viz. here (m— 2) of the conics come to coincide with the two points considered as a 
point-pair or infinitely thin conic. If the points 2 and 2 come to coincide, that is, if the 
four given points on the curve all coincide, there is no further reduction, but we have 
(4, l)=m-{-n— 6. 
66. The expressions involving a single (1) may in every case be reduced by the fore- 
going method to depend upon other expressions ; thus we have 
(3Z 
) (T, 1) 
= (' 
1) 
-2(2) , 
(2Z 
) a, 2) 
= (' 
2) 
-3(3) , 
» 
(1. 1. 1) 
= (' 
■1. 1) 
-2(2,1) , 
(Z 
) (I» 1, 2) 
= (' 
■1. 2) 
-2(2,2) - 
55 
t-H 
T— 1 
IrH 
!)=(' 
•1, 1, 
1)— 2(2, 1, 1), 
55 
(1, 3) 
=(' 
3) 
-4(4) , 
(1> 4) 
=(' 
•4) 
-5(5) , 
&c., 
where, comparing for example the equations for(Z)(l, 1, 2) and (2Z)(1, 1, 1), it will be 
observed that in the first case the contacts 1, 2 of the symbol (1, 1, 2) successively 
coalesce with the point 1, giving respectively 2(2, 2) and 3(1, 3), the exterior factor 
being in each case the barred number, whereas the second case, where the contacts 1, 1 
of the symbol (1, 1, 1) are of the same order, we do not consider each of these symbols 
separately ^thus obtaining 2(2, 1)+2(1, 2), =4(2, 1)), but the identical symbol is taken 
only once, giving 2(2, 1). Thus we have also 
(1, 1,1,1, 1)=(-1, 1, 1, l)-2(2, 1,1, 1). 
67. The value of a symbol involving (2), say the symbol (3Z)(2), is connected with 
that of y( 3Z • /) ; but as an instance of the correction which is sometimes required I 
notice the equation 
(2, 1, 1, 1)=K1, 1. l-/)-U(m-2)(m-3)+l(»-2)(»-3)+3(3, 1, 1) + 2(4,1)}, 
which I have verified by other considerations. 
68. We obtain the series of results : 
®( :0=1. 
( .-./)= 2 , 
( ://)= 4 , 
( 7 //)= 4 , 
( ////)= 
