320 PROCEEDINGS OF THE AMERICAN' ACADEMY. 



Then by a linear homogeneous transformation 



V = V 



w = w 



we secure the form 



f{u, V, w) =f(u, V, w) 



— U'" + (/7, V, T<,-)„,+i + 



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



f{u, V, w) = [i?' + p, (v, w) u"'~^ + + p,„ {v, w)'] E{Ji, V, w). (38) 



Now, in the expression 



7>^(y, 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 ^factor, there would have to be present in^" a 

 term of degree ^ m containing v or w. 

 Make in (38) the transformation 



M + ,^ ^1 iv W) = $\ 

 V = 7] >■ 



As jOi {y, w) contains no term of degree less than 2, by the considera- 

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



B. Thk Quadratic Transformation. 

 2. The transformation 



applied to $ {$, t], C) secures the form 



<J>(^, V, i) = C"^(l I = C"lr + ^^(1, V, 01 (39) 



Here the curve ^ (i, rj) = becomes |"' — 0, and so, applying the 

 Lemma of § 2 to a circle in the v^-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 1^-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 t = 0, and the point at infinity. 



