320 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Then by a linear homogeneous transformation 



u = au + (3v + yW 

 v = v 



w = w 



we secure the form 



f(u, v, w) =f(u, v, w) 



— u m + («, v, w) m+l + 



By Weierstrass's Theorem we can express this in the form 



f{u, v, w) = \u m + p, (v, w) u" 1 - 1 + + p m (v. w)-] E{u, v, w). (38) 



Now, in the exjjression 



p K (v, w), A = 1, 2, m 



there is no term of degree less than A + 1, for otherwise on account of 

 the constant term in the i£ factor, there would have to be present in^a 

 term of degree < m containing v or w. 

 Make in (38) the transformation 



u + r* Pi ( v w ) 



v 

 w 





As pi (v, w) contains no term of degree less than 2, by the considera- 

 tion above, f goes over into form (37). 



B. The Quadratic Transformation. 

 2. The transformation 



£ = i£> v — v& 



applied to <E> (£, 77, £) secures the form 



*(*, v, = *"•*(£ v, = £*[?" + £*(!, v, 01 (39) 



Here the curve <f> Q, ij) = becomes | m = 0, and so, applying the 

 Lemma of § 2 to a circle in the y^-plane however large, we have within 

 it but a finite number of singular points to treat further. But one such 

 circle is needed, for by taking it large enough we can deal with all of 

 the ^-plane outside of that circle by the transformation 



So we need to consider for further treatment only a finite number of 

 points along the line $ = 0, and the point at infinity. 





