806 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



m l being the lowest degree of any term in x t and z together, and 

 F(x u y, z) including all terms of degree higher than m^ in the two 

 variables x u z. Each coefficient p r3 (y) may be divisible by a power 

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

 in x™ is present in $ a (a^, y, z). 



By means of a transformation with non-vanishing determinant, 



x x = ax x 2 -f /?! z 2 ) 



Z = a 2 X 2 + Pi Z 2 ) 



4>j can be thrown into the form : 



*i (*i> V, z ) = $2 fa, y, z 2 ) = 

 9o(y)^ m t + qi(y)x 2 ' n - 1 z, + + q mi (y)z 2 "h + F,(x 2 , y, z 2 ) = (,) 



where q (y) =j= 0. 



Consider first the points of the circle \y\ < h at which q (y) = 0, if 

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

 lar point of <J> 2 = of order not greater than /«, and its neighborhood 



|*i|<«i |y-y*|<«, M< s 



may be chosen arbitrarily small. 

 Surround each of these points in the 

 circle \y\ = h by a circle of arbitrar- 

 ily small radius e'. 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 



#2 = y — « 

 and let <J>. 2 then be written in the form 



$2 (*2, y, * 2 ) = *2 (*2> Vi, * 2 ) = 



<A>0 2 )*2 mi + q~i{yd x ™ l ~ l z * + + q Mj (y-i)z, n \ -f F 2 (x 2 , y,, z 2 ) 



= [*."•« + nbtixt-i- 1 * + + r m Jy 2 )l. 2 '"qF(y 2 )+F 2 (x 2 ,y 2 ,z 2 ) 



= 0. (6) 



3. Apply to the function <I> 2 the quadratic transformation 



X 2 =^ x$ z 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 

 Ivobb's analysis. They appear to be indispensable. 



