( 45G ) 



ho roquiroil to toueli — or at least to intersect in two eoincidcnt 

 points — the curve ƒ three times. From the node six tangents, the 

 inflexional tangents included, can be drawn to the curve, three of 

 which (.r — a,), (.r — flo), (.k — a-.) make up tog-ethei- the curve R^; by 

 joining to each of the remaining tangents i^—ci), {.v—cq), (x — Cg) a 

 line through the node (^ — ^'i), {.r — /-o), (.r — hr.)^ counted twice, the 

 four functions thus obtained 



^1 = (•'■ -Ci){-r - b,f, n, = (,r - r,){.v - b,f, 

 R, = (.r - C3)(.r - b,f, R, = 0' - «,)(.. - r,o)(..' - «3) 



indeed satisfy all the demands, if only we take care to choose the 

 quantities h in such a way that the four functions are in involution. 

 With the aid of this condition we can eliminate the Z<'s, after which 

 still one relation remains between the a's and the c's. Hence of a 

 reducible integral five branch-points out of six may be chosen arbitrarily. 

 Reducible integrals of the kind considered here are easy to con- 

 struct if w'e observe that the four binary cubics belong to the 

 system of first polars of a binary biquadratic «■/. Among these polars 

 there are four of the form Or — c){^ — />)2, having a double point, so 

 besides R^, R^, R^ also {■>■ — r^) {.r — ^4)" belongs to the system 

 and the four quantities b are at once recognised as the roots 

 of the Hessian A*. Also cj, co. rg, c^ are the rootGi of a covariant, 

 found b\' the following consideration. The four points y, whose 

 first pohirs a,/ f/^'^ contain a double point b, are the roots of the 

 covariant '} 3 i A* — ^jf<.r*, where i and j denote the two invariants 

 of «/. The result of the elimination of y between this covariant 

 and r/^ «,/', which result is of the 12* order in .r and of the 8^'' 

 order in the coefficients «, must b(^ a covariant having the quan- 

 tities b for double roots and the quantities c for single roots. After 

 division by the square of A* a covariant of the 4"' order in the 

 coefficients a will remain, necessarily of the form Ai A* +,«,;' «A 

 the roots of which are c^, co, c^, (■4. To determine the coefficients A 

 and //, wc consider the special case 



2i 2 



a/ ^ - .,:"• {." - 1 ) , a; = - ~ (2 .H + 1). 



In this case a/ has a double ]ioint, the foui' values of /• are 

 ') Cleuscu-Lindemann. Vurh-f:iiiii/i-ii, I, p. 231. 



