124 
PROFESSOR CAYLEY ON THE CURVES 
which is 
= — (m— 4) 
31(2$ + 3 *) 2 + 110(2& + 3* )(2t + 3i ) 
+(^m 3 +1142m+3174 )(*-./) 
+ (-14w-638w-1524 )(2i + 3*) 
+ (-390m + 110# + 4272)(2r+3/) 
+ (-210m 2 -180mw + 2130m+990w-7110> + 135^ 2 
but I have not sought to further reduce this expression, not knowing the proper form 
in which to present it. 
92. The question which ought now to be considered is to determine the corrections or 
supplements which should be applied to the foregoing expressions (a), (a, b), &c., or to 
their equivalents [«], [a][^]+[«, 5], &c. in order to obtain formulae for the cases beyond 
the limits within which the present formulae are applicable ; but this I am not in a 
position to enter upon. If the extended formulae were obtained, it would of course be 
an interesting verification or application of them to deduce from them the complete 
series of expressions (1 : (2 .•.) ... (1, 1, 1, 1, 1) for the number of the conics which 
satisfy given conditions of contact with a given curve, and besides pass through the 
requisite number of given points. It will be recollected that throughout these last 
investigations, I have put De Jonquieres’^ = 0 ; that is, I have not considered the case 
of the curves C r which (among the conditions satisfied by them) have with the curve U m 
contacts of given orders at given points of the curve ; it is probable that the general 
formulae containing the number^? admit of extensions and transformations analogous 
to the formulae in which jp is put=0, but this is a question which I have not con- 
sidered. 
93. The set of equations («)=[«], (a, 5)=[«][5]-j-[«, 5], &c., considered irrespectively 
of the meaning of the symbols contained therein, gives rise to an analytical question 
which is considered in Annex No. 7. 
The question of the conics satisfying given conditions of contact is considered from 
a different point of view in my Second Memoir above referred to. 
Annex No. 1 (referred to in the notice of De Jonquieres’ memoir of 1861). — On the 
form of the equation of the curves of a series of given index. 
To obtain the general form of the equation of the curves C” of a series of the index 
N, it is to be observed that the equation of any such curve is always included in an 
equation of the order n in the coordinates, containing linearly and homogeneously 
certain parameters a, b, c . . ; this is universally the case, as we may, if we please, take 
the parameters ( a , b, c . .) to be the coefficients of the general equation of the order n \ 
but it is convenient to make use of any linear relations between these coefficients so 
as to reduce as far as possible the number of the parameters. Assume that the 
number of the parameters is 1, then in order that the curves should form a 
