WHICH SATISFY GIVEN CONDITIONS. 
107 
forms of 1, 2, and the annexed 1 indicates the additional point of intersection arising 
ipso facto from the point 1 or 2 being a cusp. Similarly, we should have the symbols 
3*1, 4zl, 5x1 ; but it is to be observed that at a cusp of the curve there is no proper 
conic having a higher contact than 2*1 ; thus if the symbol contains 3x1, or a fortiori, 
if it contain 4*1 or 5al, the number of the conics is in every case =0 ; it is thus only 
the cases 1*1 and 2x1 which need to be considered. 
71. The several modes of investigation which apply to the case of contact at a given 
ordinary point of the curve are applicable to the case of contact at a cusp : we may if 
we please employ the functional method ; we have here a functional equation of the fore- 
going form, <p(m-\-vri) — <pm= given value (that is, <p(m+W, n-\-ri, a -{-a ') — <p(m, n, «) = 
given function of (m, n, a, m!, n', a!), and the general solution is as before = Particular 
Solution -j- Constant; so that there is in each case a single arbitrary constant to be deter- 
mined by special considerations. The determination of the constant is in some instances 
conveniently effected by means of the case of the cuspidal cubic : see Annexes Nos. 4 and 5. 
The formation of the functional equation itself is similar to that in the corresponding 
case where the given point on the curve is an ordinary point. For example, we have 
(2Z)(1, 1, 1W -(1,1,1),= (1,TU1)„ = »'(1, 1 • 1 /) m 
+(I)„(1, I)* +1K-»')(I:) W 
+ (m'ri— |a')(I • /) m 
+iK -W)(I//) m , 
and we may herein simply change 1 into 1*1. Writing successively 2Z=( :),(•/) and 
(//), we find 
(Txl, 1, l:) m+ml -(: ) m =ri( w+2m-3)+m , (2w+2m-6)+0' 2 — |^')l+(mV-|a')2+(- 
( ‘ ’ /) m =n'(2n-\-4m— 5)-\-rri(4n-\-4m— 5)-\-(^n' 2 — ^n')2-\-(m'n' — §a')4+(- 
( //) m+m > — ( / /) m —ri(4n-{-4m— 6) + m!(4n+2m— 3) + (\ri 2 —±ri) 4 + (m'ri — |a')4+ (- 
which only differ from the corresponding expressions with 1 in that they contain 
n-\-2m— 3, 2n-\-4m— 6, 4n-\-4m— G, 4m-\-2n— 3 
in place of 
n-\-2m— 2, 2n-\-4m— 4, 4n-\-4m— 4, 4m-\-2n— 2 
repectively, and they lead to the expressions for (1*1, 1,1:), &c., the arbitrary con- 
stant being in each case properly determined. 
= 1 , 
= 2 , 
=4, 
= 4, 
= 2 ; 
K 
72. We have 
( ^( :: ) 
(•••/) 
(:// ) 
( 7 //) 
( I III) 
MDCCCLXVIII. 
\rri 2 —\m!) 
pri*-%rri) 
pri 2 —\iri) 
