ON ALGEBllAIC HYPERCONICAL CONNEXES IN SV'ACE, ETC. 485 



3. The number of essential parameters in the equation (1) 

 will next be determined. The general polynomial, homoge- 

 neous of degree m in [x^ . . . x^) and of degree n in (//o . . . y^), 



contains — -- , homogeneous coefficients. If the poly- 



m\r\ n\r\ i j 



nomial, equated to zero, determines an hyperconical connex, 

 then ali its derivatives with respect to {i/Q..,yr) of order n — 1 

 vanish identically when (y) coincides with {x). The number of 



(nA-r 1)' 



these derivatives is , ' ., — ^. The condition that any one of 

 (w— l)!r! -^ 



them vanish identically when (u) coincides with (x) is >"~-~^^ 



linear conditions on the coefficients of the polynomial. If these 

 equations of condition are ali independent, it will follow that the 



required number of essential parameters is ' , ^"''"^ ii — 



mi ri ni ri 



_ {m+r-\-l)l (n+r-1! 



(w4-l)!r!(n-l)!r! ' 

 To show that the "' '/,', '^ — '~~ ,' linear equations of 



(m-|-lj! ri (n — 1); ri ^ 



condition are independent, it will be sufficient to show that it 



m n 



is possible to find a polynomial G{xq . . .x^; y^ . . . y,) for which 

 ali the derivatives with respect to [yo . . . yr) of order n — 1 

 vanish identically when (//) coincides with [x), except an arbitrarily 



chosen one, ^-77^—,^ — r^rr 1 which reduces to Cx(fiox.^\,.Xr'^>', 



where C is an arbitrary, finite Constant and ao,ai...a,.; Po-Pi---Pr 

 are positive integers, or zero, such that aQ-\-(Xi-{-...-\-a,.^=n — -1 



and Po+Pi+...+ P.= ^w+l- 



Suppose, first, that w=:l. Let the notation be chosen so 

 that 3o ^ 0. Tlien G ^ Cyoa^o'^*'"''^/' — ^r^'' is an expression of 

 the required type. 



The proof for n > 1 will be by induction, the existence of 

 expressions of the required type being supposed established 

 when the degree in {y^ ... y^) is less than n. 



Let the notation be chosen so that Po =*= 0. Let k — 1 be 

 the number of the quantities a^, Og ... a,, which are different from 



m — 1 n — 1 



zero. Let y^XQ . . . Xr; yo ■ • • l/r) be a polynomial which satisfies 

 the conditions that (1) if o-j= 0, (/j = 0, (2) if a^ =p 0, gj is ho- 

 mogeneous of degree m — 1 in {xq . . . x^) and of degree n — 1 



Atti della R. Accademia — Voi. XLVI. 31 



