WHICH SATISFY GIVEN CONDITIONS. 
83 
19. If the conic has simply to touch a given curve of the order m, and class then 
the order of the condition (or number of the conics which satisfy the condition, and be- 
sides pass through four given points) is equal to the order of the contact-locus, that is, 
it is =n i -\-2m 1 . If the conic has also to touch a second given curve of the order m 2 and 
class n 2 , then the order of the twofold condition (or number of the conics which satisfy 
the twofold condition, and besides pass through three given points) is equal to the order 
of the intersection or common locus of the two contact-loci ; and these being of the 
orders n i -\-2m l and n 2 -\-2m 2 respectively, the order of the intersection and therefore that 
of the twofold condition is ■={n l -\-2m l ){n 2 -\-2m^). But in the next succeeding case it 
becomes necessary to take account of the singular locus. 
20. If the conic has to touch three given curves of the order and class (m 15 nj, 
( m 2 , n 2 ), (m 3 , n 3 ) respectively, we have here three contact-loci of the orders (n l -\-2m ] ), 
n 2 -\-2m 2 , n 3 -\-2m 3 respectively; these intersect in a threefold locus, but since each of 
the contact-loci passes through the threefold Bipoint-locus, this is part of the intersec- 
tion of the three contact-loci ; and not only so, but inasmuch as they pass through the 
Bipoint-locus m 2 , m 3 times respectively, the Bipoint-locus must be counted m{m 2 m 3 
times, and its order being =4, the intersection of the contact-locus is made up of the 
Bipoint reckoning as a threefold locus of the order 4m 1 m 2 m 3 , and of a residual threefold 
locus of the order 
(n l -f- 2m i )(n 2 -{-2m 2 )(n 3 + 2m 3 ) — 4 m{m 2 m 3 , 
=w 1 w 2 w 3 +2(w,w 2 m 3 +&c.) +4(wpw 2 m 3 -f &c.) + 4m 1 m 2 w2 3 ; 
and the order of the threefold condition (or number of the conics which touch the three 
given curves, and besides pass through two given points) is equal to the order of the 
residual threefold locus, and has therefore the value just mentioned. 
21. In going on to the cases of the conics touching four or five given curves, the same 
principles are applicable ; the contact-loci have the Bipoint (a certain number of times 
repeated) as a common threefold locus, and they besides intersect in a residual fourfold 
or (as the case is) fivefold locus, and the order of the condition is equal to the order of 
this residual locus ; but the determination of the order of the residual locus presents 
the difficulties alluded to, ante , No. 10. I do not at present further examine these cases, 
nor the cases of the conics which have with a given curve or curves contacts of the 
second or any higher order, or more than a single contact with the same given curve. 
22. The equation of the conic has been in all that precedes considered as containing 
the six parameters (a, b , c, f, g, h ) ; but if the question as originally stated relates 
only to a class of conics the equation whereof contains linearly 2, 3, 4, or 5 parameters, 
or if, reducing the equation by means of any of the given conditions, it can be brought to 
the form in question, then in the latter case we may employ the equation in such re- 
duced form, attending only to the remaining conditions ; and in either case we have the 
equation of a conic containing linearly 2, 3, 4, or 5 parameters, which parameters are 
taken as the coordinates of a point in 1-, 2-, 3-, or 4-dimensional space, and the discussion 
relates to loci in such dimensional space. This is in fact what is done in Annex No. 2 
MDCCCLXVIII. 
o 
