BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 293 



where g a has a term in ^ free from Vl aud £, since a a — a a , £ 0. So 

 there are near the point (0, 0, 0) ^ values of £ a satisfying the equation 

 g a — for every pair of values of ^ aud £ in the neighborhood of the 

 point r\ x = 0, £ = 0. Now, for any such set of values of c , rj u £, 

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

 a corresponding set of values of £, rj, £ satisfying the equation $ (f, 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 IT (£ — a^ ", we get (s — 1) other equations of form 



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

 form (4). 



No two points (£ v , £), (£ f , v >, £') of T (distinct from (0, 0, 0)), de- 

 rived from points (£ ffl rj v Q (^„ %, £2) lying respectively in the neigh- 

 borhoods of the singularities which are given by two distinct equations 



9, = °, <J°' = 0, 



can be the same. For suppose 



*=& = It (4 + a,) = £ 2 U a , + a,,) 



£ = £' = Ci = £ 2 



Then we must have 



4 + a <r = £r' + <V' 



4 - £ 



'cr' 



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

 ciently small, we can insure that the difference £ v — £ a , is less in abso- 

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

 that, if the equation g = 0, regarded as an equation in £ , has equal 

 roots for all values of ij 1} £ in the neighborhood of the point -q x = 0, 

 £ = 0, the equation <S> = must also have equal roots at the corre- 

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

 g = has near the point (0, 0, 0) p values of £ , in general distinct, 



t 

 for each pair of values of rji and £, aud as 2 /j. = m, the collection of 



equations 



g a = , o- = 1 , 2, 8 , 



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



