( 819 ) 



To determine a particiilai' solution T{m) = .i"» we have 

 ,,^ _ (,, _ 2) (« + 2).. + (« - 2)^ {n + 1) = 0, 

 therefore 



,y zz: ir — n — 2 Or A' z=: /t — 2. 



Consequeiilly tlie general solution is 



r{m) = c,(n' - n - 2)"' + .,(« - 2)"'. 

 To determine the constants c\ and c\ I substitute in (4) the known 

 values (M + l) of t(1) and (u + 1) (/t'^ — 4) of t(2). 

 Now 



/( 4- 1 = c^ {>,^ _ ,, — 2) + c, (n — 2), 

 (ir — 4)(« + 1) = 0, («■- — « — 2)-^ -(- c-, (m — 2)^ 

 Finally we find by elimination of c\ and c, 



T(m) = (« + 1) {n — 2)"'-i "— ïl^! .... (7) 



n 



From (1) and (2) ensues 



« (m) — i? (»») =: - 2 <« (m — 1) — ,? (w _ 1)> , 



so 



« (m) - iJ (m) = (- 2)"'-> \a (1) - ,? (1)| = - (- 2)"' . . (8) 

 Making use of (3) and (7) we now find 



//- (t (;/i) = (« — 2)"'-l |(h + 1)"'+' — -In + Ij — {»■' — 1) (— 2)'« . (9) 

 ,,'^ ,i(»i) = {n — 2)'«-l |(n + 1)'»+I — 2« + Ij + (— 2)»' .... (10) 



§ 4. For in = 2 we (liul « (2) = ///" -f- // — 9 ; as ^1 is inflection 

 for three curves c„, tiiere are tlierefore (n^ ~\- n — 12) curves on 

 which A coincides with its second tangential point. From this ensues 

 the wellknown result that A is point of contact of (n -f- 4) (n — 8) 

 double tangents. 



In a former paper ') 1 have brought into connection the locus of 

 the points of contact D of the double tangents with the locus of the 

 points ir in which a r" is cut l)y ils double tangents. To determine, 

 how often a point D coincides with one of its tangential points IF 

 I consider the correspondence of the rays c? = OD and iv := Ojr 

 which tlie correspondence {D, W) forms in a pencil with vertex 0. 



As the curves (D) and ( IF) are of orders (n — 3) (2»" -|- 5n — 6) 

 and i (// — 4) (n — 3) (5h^ -\- 5;i — 6), to each ray (7 correspond 

 (/i — 4) {n — 3) (2«^ -|- 5ii — 6) rays ;/' and to each ray w correspond 

 {n — 4) {ii. — 3) (5»" -|- 5/i — ()) rays (/. 



'j 'On liiii'ar systems of algeln-aic plane curves." Proc. April ii2 1905, Vol. VII 

 (a), P- 711- 



