BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 293 



where g^ has a term in ^'^'^ free from 7/1 and ^, since a^ — a^, ^ 0. So 

 there are near the point (0, 0, 0) /i,^ values off satisfying the equation 

 g^ = for every pair of values of 771 and ^ in the neighborhood of the 

 point 7;i = 0, ^ = 0. Now, for any such set of values of $ , t^i, ^, 

 different from the set (0, 0, 0), satisfying the equation g — 0, there is 

 a corresponding set of values of f, 77, ( satisfying the equation 4> ($, rj, ^) 

 = 0, their coordinates being connected by the relations (2), (6), and (6'), 

 which are equivalent to the required relation (4). Also by considering 



s _ 



the other factors of 11 ($ — a^)'^'^, we get (s — 1) other equations of form 



(7), the corresponding coordinates being connected by relations of 

 form (4). 



No two points (f, rj, ^), (|', rj', tj) of T (distinct from (0, 0, 0)), de- 

 rived from points (f^, 77^, ^j) (f^„ tj^, ^2) lying respectively in the neigh- 

 borhoods of the singularities which are given by two distinct equations 



can be the same. For suppose 



^ = 4^' = ^: (f<, + a^) = 4 (4' + «a') 



^ == C = Ci =C. 



Then we must have 



4 + a<r = 4' + «<r'' 



and, by taking the neighborhoods of the singularities in question suffi- 

 ciently small, we can insure that the difference ^^ — ^^, is less in abso- 

 lute value than the difference u , — a . In a similar manner it is shown 

 that, if the equation y = 0, regarded as an equation in f , has equal 

 roots for all values of 771, ^ in the neighborhood of the point 771 = 0, 

 ^ = 0, the equation 4> = must also have equal roots at the corre- 

 sponding points, and this case has been excluded. So as each equation 

 ^r = has near the point (0, 0, 0) ^ values of ^ ? in general distinct, 



for each pair of values of 7/1 and ^, and as 2 // = m, the collection of 



equations 



9^ = ^^ 0-= 1, 2, s, 



has within sufficiently small limits as many different roots as the equa- 



