102 
PKOPESSOR CAYLEY ON THE CUKVES 
Again, 
(2, 1, 1, l) m+ml —(2, 1, I, l) m — (2, 1, 1, l) w = (2, 1, lUl) m , + (2, 1, lUl) m 
+(2, 1).(1, l)w+(2, 1U1 ). 
+ (2) m (l, 1, l) m ,+(2) w (l, 1, l) ro , 
and 
5400 + 9<2 + 18c=0, 
2280 + 7a+12c=0, 
1680 + 6a+10c=0, 
satisfied by <2=1 320, c= — 960 ; and finally, 
(1, 1, 1, 1, l) m+ w-(l, 1, 1, 1, l) m -(l, 1, 1, 1, l) m ,= ’ (1, 1, 1, l) m (l)w+(l, 1, 1, 1U1)™ 
+ (l,l,l) ro (l,l) w +(l,l,lUl,l) m , 
and 
— 30618 + 90(2+180^=0, 
— 14094+ 70a+120<?=0, 
— 10692+60<z+120<?=0, 
satisfied by 10«=6318, 10c=4860, that is, a—— — 5 —, c=486. 
62. The contacts of a conic with a given curve which have been thus far considered 
are contacts at unascertained points of the curve; but a conic may have with the given 
curve at a given point thereof a contact of the first order, the condition will be denoted 
by (2) ; or a contact of the second order, the condition will be denoted by (3), and so on. 
It is to be observed that the conditions (2), (3), &c. are sibireciprocal, the contact at a 
given point of the curve is the same thing as contact with a given tangent of the curve ; 
but if we write (1) to denote the condition of passing through a given point of the 
curve, this is not the same thing as the condition of touching a given tangent of the 
curve ; and this last condition, if it were necessary to deal with it, might be denoted by (1). 
But I attend only to the condition (1). The expressions for the number of conics which 
satisfy such conditions as (1), (2), &c. are obtainable in several ways. 
63. (1°) When the total number of conditions is 4, the question may be solved by 
Zeuthen’s method, viz. by determining the line-pairs and point-pairs of the system 4Z, 
with the proper numerical coefficients, and thence deducing the values of the character- 
istics (4Z • ) and (4Z /). A few cases are in fact thus solved in Zeuthen’s work. 
64. (2°) By the foregoing functional method. It is to be observed that there is a 
difference in the form of the functional equation, and that the general solution is always 
given in the form, Particular Solution + Constant , so that there is only a single con- 
stant to be determined by special considerations. To take the simplest example, let it 
be required to find the number of the conics (3Z) (I, 1): writing for shortness in place 
hereof (1, 1), or (in order to mark the curve (to) to which the symbol has reference) 
(1, l) m , let the curve (in) be the aggregate of the curves (to) and (to'). Regarding the 
point 1 as a given point on the curve (to), that is, an arbitrary point in regard to the 
