406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
199 
contact at each of the points of intersection of the given curve with the coincident 
line-pair regarded as a single line, that is, in the case of a given curve of the m-th 
order, m special contacts, the Bipoint-locus is a multiple curve on the corresponding 
contact-locus. 
19. If the conic has simply to touch a given curve of the order m x and class n 1} then 
the order of the condition (or number of the conics which satisfy the condition, and 
besides pass through four given points) is equal to the order of the contact-locus, that 
is, it is =n 1 + 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 x + 2n\ and n 2 + 2m 2 respectively, the order of the intersection 
and therefore that of the twofold condition is = (n x + 2m 1 ) (n. 2 + 2m 2 ). 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 1 , %), 
(m 2 , n 2 ), (m 3 , n 3 ) respectively, we have here three contact-loci of the orders n 1 + 2m 1 , 
n 2 +2m 3 , n 3 + 2vi 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 intersection 
of the three contact-loci; and not only so, but inasmuch as they pass through the 
Bipoint-locus Wi, m 2 , m 3 times respectively, the Bipoint-locus must be counted m l 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 w 2 m 3 , and of a residual three 
fold locus of the order 
(n x + 2m x ) (« 2 + 2m 2 ) (?? 3 + 2m 3 ) - 4w 1 m 2 m 3 , 
= n^ris + 2 {n l n 3 m 3 + &c.) + 4 (n l m i m 3 + &c.) 4- 4?n 1 m 2 m 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 con 
taining 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