in relation to the theory of point-groups. 475 



and if all these a equations are independent, that is, unless for 

 some set of coefficients 



i/r [xr, 2/r) = (r = niu — « + 1, ... mu). 



This is the theorem of Cayley and Bacharach, that a curve of 

 degree r{<m+n — 2), which passes through all but 



^(m + n — r — l)(m + n — r—2) 



of the mn intersections of two curves of degrees m, n, passes 

 through the excepted intersections also unless these excepted 

 points lie on a curve of degree m + n — r — S. There is no ex- 

 ception when r = m + n — S. 



Similarly in three dimensions, a surface of degree ni + n+p — 4!, 

 which passes through all but one of the intersections of three 

 surfaces of degrees m, ?i, p, passes through all, and a surface of 

 lower degree r must pass through all the intersections if it 

 passes through all but 



^(m + n + p —1 — 1) (m + n + p — r— 2)(m+ n +p — r — 3), 



unless these excepted points lie on a surface of degree 



m + n + p — 1 — 4. 



Bacharach has further noticed that the lowest value of /8, 

 such that an r''^ curve can pass through mn — /8 of the inter- 

 section of u, V, say A, without passing through the rest, say B, 

 is m + n — r—1, and that in such a case the points B are collinear. 



For if i/r is any (m + n — r — 3)'° we have 



2</>^/r/J=0, 



the summation being over the /S points B, and thus any i^ 

 through all but one of these passes through the other. If 



^ = m -\- n — 1 — 2, 



take ■x/r to consist of straight lines drawn from an arbitrary origin 

 to all but one of the points B: this composite curve cannot always 

 pass through the excepted B point, if the B points are distinct. 

 Thus /3 cannot be less than m + n — r — 1. 



If ^ = m + n — r—l, suppose B^, B^, B^ to be three of the 

 points B, not in a straight line, and take i/r to consist of the line 

 BiBo and lines joining B^, B^, B^ ... to an arbitrary origin: this 

 will not always pass through B3. Hence all the points B must be 

 collinear. 



Also fi can have this value. For take 



U ^ Xjyyi^i Jm+n—r—iJr—n+1 > 

 '^ ^^ •^Jn—i Jm+n^r—ijr—m+i) 



