312 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



and all the resulting linear factors would have to vanish when 



£ = 0, 77 = 0, 



and so not contain £. 



Also by a linear homogeneous transformation in £ and rj we can se- 

 cure the presence of terms in £"* and rf 1 , and in such case every linear 

 factor of <£ (f, rj), which here is (£, rj) m 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 



f = £i£i 7 = 7i£> 



and we have 



*(£, r;, o = r"[(^ >?o m + £&, ti, i) m+ i + ] 



= r[&, 7i)»K«A(^,7i» 0] as) 



= r*i(ii,7i, 0- 



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 

 ^(iu 7u £) and as all terms of this contain £, when we put the expres- 

 sion in the form 



*, Hi, VI, = (*1. 7l, Om + (*„ 71, 0-+1 + ( 16 ) 



we will secure reduction by another quadratic transformation unless 

 ($v 7i» Om is tne product of w linear factors with a common line of 

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

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

 factors of (£ 1} rji) m . Also the common point of intersection of the lines 



