( 818 ) 



To deterniine the fuiu-liuns r[m), a{m) and ji(»i) 1 shall make use 

 of an auxiliary curve already used by Weyk, which might be called 

 the antitnn(jential curve of A. It contains tlie groups of a {71 — I)- — 2 

 points ^1— 1_ having A as tangential point; so it passes three times 

 through ^4 and once through all points B. So it has (2»" — n) 

 points in common with any c", from which it is evident that it is of 

 order (2h — 1). 



^ 2. The {m — 1)* tangential curve (^'«-') of A is cut by the 

 antitangential curve (^~') of A, save in the base points, in the points 

 A"'~^' having A as tangential ])oint. Their number amounts to three 

 less than the number of tangents which T,,, has in A, so a{m) — 3; 

 for, on the three c" which have in .1 an iutlcction .1 coincides with 

 one of its vh"' tangential points. 



The three iiitlectional tangents being also tangents of the curve 

 {A~'), the tangential curve (^1'""') and the antitangential curve 

 (J.~') have 3«(j/i — l)-)-3 pohits in common in A. In each base- 

 point B lie [i^iii — 1) points of intersection. So 



(2w — 1) T {m — 1) = » (?w) 4 3ft (w — 1) + («' — l)^(m — 1) . . (1) 



A .second relation is found by noticing that (/]'"—') has with the 

 antitangential curve of B, save the basis, the l3{m) points in common 

 for which B is an ?/i''Mangential point. In B lie 3|J (/», — 1) points 

 of inter.section, a {in — I) points of intersection lie in A, mm — 1) 

 in each of the other basepoinls. So 



(2» - 1) r(m - 1) = ,i{,n) + a(m - 1) + ("' H 1) /:?('" - 1) • (2) 

 With any r' the locus 2'„ has, save the i)asis, only the (n — 2)'« 

 points ^1'^'") in common ; so 



n T{m} =-- a(in) + {u' — 1) /i(»t) + {n — 2)'" . . . (3) 



§ 3. To tind a iiomogeneous equation of finite differences for the 

 determination of t(7//) I eliminate from the three obtained relations 

 the quantities a'm) and |5(/h), and 1 find 

 nr{m) = »'{2)i — l)T(m - 1) — (m' -U 2)ift(w— 1) -f (>,.- — I )i^m - 1 )] + (« - 2)"'. 



Here the expression within braces can be replaced on account 



of (3) by mim — 1) — (»• — 2)'"-'. Then 



t(m) = («' — « — 2) t(7/i — 1) + (m + !)(« — 2)"' 1 . . (4) 

 So 



T(m — 1) = («' — II — 2) t(?« — 2) + (n + 1)(m — 2)'"-- . (5) 

 Equations (4) and (5) finally furjiish 

 T(m} — (n — 2)(« + 2) T(m — 1) + (« — 2)^(« + 1) r{m — 2) = (6) 



