288 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 an analytic function. 



The symbol E (x,y, z, ) will always represent a function which 



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



4= 0. If written with a subscript, as E r (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 (z, 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) = '0 



can be transformed to the form 



<*> (£ V, = (£ r), 0™ + (6 V, Om+i + - o 



where 



1) m > 2, 



2) the polynomial 



(£, v , i) m = *(£, v) 



contains the term £ m , 



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

 factors of <£ (£, r/) cut the line at infinity shall be distinct from each 

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



To do this, we first make the transformation 



x = u + a, = v + b , z = w + c , 



thus obtaining 



F(x, y, z ) =f(u, v, w) = (u, v, w) m + (u, v, w) m+l + 



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

 a linear homogeneous transformation with non-vanishing determinant, 



U = Ox | + /?! 7} + yi £ J 



v = a,£ + (3 2V + y,C> (1) 



w = a 3 £ + & rj + y 3 £ ) 

 with the result : 



/(U, V, W) = $ ($, Tj, = (6 ">?, Dm + (£ ??> Qm+1 + = . 



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

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



