WHICH SATISFY GIVEN CONDITIONS. 
103 
curve (m'), we have thus the equation 
• (I, i) m+m ,-(I, i) m =( ■ 1U 
where the right-hand side is known ; and so in general the form of the functional equa- 
tion is always <p(m + w!) — <p(m) = given value , that is, 
n-\-n\ a-\-a')—<p(m, n, a) = given function of (m, n, a, ml, n', a!); 
whence, as stated, the general solution is Particular Solution + Constant. In the case 
in hand, taking successively (3Z) = (.\), ( : / ), ( • //), and (///), we have in the first of 
these cases 
(I, l) m =n'+2m', 
whence (1, Y) m =-n-\-2m-\- const. =(T, 1)(.‘.) ; and the value of the constant being in 
any way ascertained to be = — 2, we have (1, 1) (.\)=wfi-2m — 2 ; and the like for the 
other three cases. 
65. (3°) The expressions for the number of conics which satisfy such conditions as 
(1), (2), &c. are deducible with more or less facility from the corresponding expressions 
wherein (1), (2), &c. are replaced by (•),(: ), &c. ; thus from ( : : Y)*=n-{-2m we deduce 
(•••I, l) = (::/)-2(.-.2)=rc+2m-2, 
viz. if one of the four arbitrary points of ( : : / ) becomes a point on the curve, then the 
condition (::/) is satisfied specially by the conic (.’.2) which passes through the 
remaining three points and touches the curve at the point in question ; 2 of the conics 
(: :/) coincide with the conic in question. We have thus a reduction 2(.\ 2), =2, and 
the number of the conics (.‘.I, 1) is =n-\-2m— 2. Similarly, we have the system 
(.-. 1, 1 )=n-\-2m— 2, 
(: 1,1,1 )=.n-\-2m— 4, 
(• I, T, I, l)=n+2m-6, 
(I, I", I, I, l)=w+ 2m-8. 
Again, two or even three of the given points on the curve may come together without any 
reduction being thereby caused, that is, we have 
( : 2, 1 ) =w+2to— 4, 
(•2,1,1, )=(-!, 1 )=n+2m-Q, 
( 2, T, 1, 1)=( 3, 1, l)=rc+2m-8; 
but if the four points on the curve coincide in pairs, or, what is the same thing, if in 
(2, 1, 1, 1) the points 1 and 1 come to coincide, then there is a special reduction, and 
we have 
(2, 2, l)=w-f-2ra— 8 [— (m— 2)]=m+w— 6, 
* I write indifferently (1) (: :), (1 : :) or (: : 1) ; and so in other cases. 
