PROFESSOR CHRYSTAL ON THE HESSIAN. 649 



where from the nature of the process employed we are sure that there is no 

 redundant factor. 



Now af n or x mv is a factor in (3) according as ^n< or >mv, i.e., the number 

 of zero roots of (3) is the least of the two numbers ^n mv. 



Hence denoting for shortness the least of the two jxn mv by l{^n, mv), we 

 have the following simple theorem. 



The number of intersections of x m — y n = and of— j/" = at the point 

 x = y — 0, is 



l(iu.n, m v) . 



4. By means of the Newton-Cramer rule we can, as far as points near 

 x = # = are concerned, replace U and V by 



U' = (of i - A x y">) (x - A 2 y"i) . . . . 

 V = (aft - B, yi) (a?* — B 2 y v *) ..... 



where the factors in U' will in general be all different, those in V all different, 

 and no factor common to XT' and V. 



In this case we can at once find the number UV. We have, in fact, 



UV=U'V' = l(m 1 i/j , n x fj. x ) + l(m x v 2> n x m 2 ) + ■ • • • 

 +l(m 9 v t , « 2 mj+lfat i/j, n 2 m 2 )+ • • ■ • 

 4- 



This result still holds when factors are repeated in U' or in V; but when there 

 are factors common to U' and V there is a modification, as will be shown 

 presently. 



5. Before proceeding farther, let us apply the above principles to one or 

 two examples. 



Ex. 1 . U = x 3 u 5 + xht 7 + u 10 — 



V x i v 1 + x i c 3 + v 7 =0. 



The Newton-Cramer diagrams for U and V show at once that (omitting 

 constant coefficients as irrelevant to the issue) we may write 



.-. UV = 5x 1 + 5x4+3x1 + 3x6=46, 

 the same result as before. 



Ex. 2. U = x 3 u 5 + x u s + u 10 



V = xH x + x 3 v 3 + v 7 . 

 Here it is easily shown that we may write 



U = u 6 (x 2 + y 3 )(x + y*) 

 YELv^x' + y*); 



.: UV = ox 1 + 5x4 + 2x1 + 2x6 + 1x1 +1x6 = 46. 



EX.3. U = %* Uk-m + Uk-r 



V — of v K _^ + V K+p . 

 VOL. XXXII. PART III. 5 R 



