288 PROCEEDINGS OF THE AMERICAN ACADEMY. 



A functional sign expressed by means of a letter will always represent 

 an analytic function. 



Tiie symbol E {x, y, z, ) will always represent a function which 



is analytic at the point (0, 0, ) and for which ^ (0, 0, ) 



^ 0. If written with a subscript, as E^ {x, y, z, ) it represents a 



particular function of the class ; if without a subscript, it represents a 



general function of the class ; so that two functions E {x, y, z, ) 



both expressed by the same symbol, need not be equal to each other. 



B. — The Transformations. 



3. The equation 



F(x,y,z) = Q 



can be transformed to the form 



^ (^, -n, = a, j7. 0». + (^, V, 0«.+i + - 



where 



1) m > 2, 



2) the polynomial 



contains the term ^'", 



a, ri, i)„. = c^ a, v) 



3) the points in which the curves corresponding to the irreducible 

 factors of ff> (f, r}) cut the line at infinity shall be distinct from each 

 other and from the point in which the line f = cuts that line. 



To do this, we first make the transformation 



X = U + a, :=! V + b , Z T=: w -\- c , 



thus obtaining 



^(^, I/, z ) =/(«, '', ■?<'•) = (u, v, w),„ + {u, V, w),„+y + 



Here, m > 2, the singularity now being at the origin. Next we make 

 a linear homogeneous transformation with non-vanishing determinant, 



V = a.i+ /3.r, + y.^V (1) 



with the result : 



f(u, V, w) = <!> ($, rj, = (t\ V, 0,„ + (^, V, 0,«+i + = . 



For this equation, conditions 2) and 3) can be secured, as is readily seen 

 by a proper choice of the coefficients in transformation (1). 



