806 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



mi being the lowest degree of any term in Xi and z together, and 

 F{xi^y^ z) including all terms of degree higher than m^ in the two 

 variables Xi, z. Each coefficient p„ {v) ™ay be divisible by a power 

 of y, y. In that case, however, m,i must be less than w, for the term 

 in Xi'" is present in 4>i (xj, y, z). 



By means of a transformation with non-vanishing determinant, 



$1 can be thrown into the form : 



*i (xi, y, z) — <I>2 (xo, y, 20) = 



9o(y)a:2"'' + qi(y)xr-'^-2 + + g„n(.i/)-2"^ + ^2(^2, y, ^2) = (ri) 



where ^o (y) ^ 0. 



Consider first the points of the circle \y\ < h at which ^o (y) =■ ^5 if 

 such exist. Each one of these points y^, (i = 1, 2, k) is a singu- 

 lar point of ^2 = of order not greater than m, and its neighborhood 



l^il<f' \y-yi\<^> \^\<^ 



may be chosen arbitrarily small. 

 Surround each of these points in the 

 circle \y\ = h hj A circle of arbitrar- 

 ily small radius c'. We now proceed 

 to consider the region about an arbi- 

 trary point a of the circle \y\ < h not 

 lying in any of the regions just cut 

 out. Let 



yi^y — a 



and let 4>2 then be written in the form 



*2 (a;25 y^ ^2) = ^2 (a^2, yz, ^2) = 



70(^2)3^2'"' + 9i (^2) a^2"' ~' 2:2 -f + q„,^ (^2)^2"'' + ^2_0^2, ^2, ^2) 



= [xs'"! -F ri(y.^x.'"-^z.^ + -f r„.^ (3^2) ^2'"^ ^ (^^2) + ^2 (•-'^2,^2, ^2) 



= 0. ' (,6) 



3. Apply to the function ^2 the quadratic transformation 



X2 — 'l'3'^2- 



* Here, for the first time, a quadratic transformation of the type that trans- 

 forms but a single variable is employed. Such transformations do not occur in 

 Kobb's analysis. They appear to be indispensable. 



