312 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



and all the resulting linear factors would have to vanish when 



^ = 0, ^ = 0, 



and so not contain ^. 



Also by a linear homogeneous transformation in ^ and r; we can se- 

 cure the presence of terms in ^ and rp, and in such case every linear 

 factor of <f) (f, rj), which here is (^, jf)^ itself, will contain ^ and thus 

 secure condition 3) of § 1, 3. 



B. — Quadratic Transformations. 



3. The succession of surfaces and corresponding quadratic transfor- 

 mations which are applied to the new singular points as found, so long 

 as they do not reduce the degree, can be written in the form 



(14) 



Apply to the surface (13) the transformation 



t — 



= ^U, 



V — ViC, 



and we have 



^{$, r), = C"[{^i, ViU + U^u Vi, l)»-+i + ] 



^S"[($l,Vl)mH^(^l,Vi,0'] (15) 



As we assume the transformation does not reduce the degree of the 

 singular point, there can be no term of degree less than m in the part 

 C^iii) Vi^ ^^^ ^^ ^^^ terms of this contain ^, when we put the expres- 

 sion in the form 



^1 (^1, ^1, = (il, VV Om + (^1, m, 0,„+l + (16) 



we will secure reduction by another quadratic transformation unless 

 (^]) Vii Om is the product of m linear factors with a common line of 

 intersection. In this case the factors cannot be all equal, for then 

 (^i> ^15 0)m would have its linear factors all equal, but these are the 

 factors of (|i, 171)^. Also the common point of intersection of the lines 



