484 e. H. SISAM 



Each of the expressions - --r — is hrmogeneous 



of degree m — 1 in the coordinates of [x) and of degree n — 1 

 in those of iy). If n = 1, the theorem is therefore established. 

 The proof for w > 1 will be by induction , the truth of the 

 theorem being assumed for any hyperconical connex, the degree 

 of whose equation in {yo • ■ • yr) is less than n. 



Ali the derivatives of order n — 2 of each of the functions 



~^^ with respect to Wn, Vi • • • (/<■» vanish identically 



òyiòxj òyjòxi 



when (y) coincides with (a?). For, let . ., . „ — s^rhr^^ ^r>r'] 



^•^^ ^ ^ òijo^ody^f^t... òyr^'-\òyibxj òyjòxil 



be such a derivative for one of these functions. By hypo- 



SI» — 1 El 



thesis: —^ — ^ „ ,i . ^ =0 if (//) coincides with {x). Hence 

 ' ' reduces to 



'■)yu"<'f)i/i°'- ^y>-^'- \ ^yiàxj ì dyo^o...dyi°ì-^^... hyf^j-^^... òyr^^ 



when {y) coincides with [x). By similar reasoning, it is seen 



that -"-^ reduces to the same expression, and 



Ò^F Ò^F 



therefore that ' ,^ ^f f ; , '^f''' ^ - = 0, when (.//) coincides 



with (x). Hence -'^ ^^r:^ = is the equation of an hy- 



perconical connex of degree n — 1 in (2/0...//^) and of degree in — 1 

 in {xq ... Xr). Hence, if m > n, 



(7) r^^ ". = 9iJ {Xo...Xr; Poi... Pr-l r) 



^ ' òyiòxj òyjòxi 



and, if m<in, 



(T) -^ ^H.0. 



Substituting from (7) or (7') into (6), we fìnd that, if m^n, 



(8) F{xo . . .x,.;yo..- yr)^f{xo. . • x^; Poi • • -Pr-ir) 



and, if m<^H, F=0. The theorem is therefore established 

 if n is any positive integer. 



