378 



received approximate value of any single incommensurable root of it 

 satisfies those conditions. And we therefore think that the indiscri- 

 minate rejection of roots involving an imaginary element, regardless of 

 the influence of this element on the coefficients of the equation, is un- 

 justifiable, and, moreover, inconsistent, where approximate values only 

 of the real roots — that is to say, exact values of the roots of only ap- 

 proximate equations — are received. 



Some remarks on this subject will be found in Peacock's paper, in 

 the " Report of the Third Meeting of the British Association," p. 349. 



There is, however, one case of equal roots which deserves special 

 notice — namely, the case in which the roots are all equal imaginary 

 pairs. 



Let the equation be one of the fourth degree, made up of two equal 

 quadratic factors {X) each having a pair of imaginary roots : the equa- 

 tion is then XX = 0. Taking the successive limiting equations (or 

 differentiating), and remembering that X being of the second degree, 

 we must have, at the third step of the operation, X'" = 0, the limiting 

 equations will be 



2 XX' = 0 [11 



2X'X f + 2XX" = 0 [2 j 



AX'X" + 2X'X" = 0, 



that is, 6X'X" = 0 [3] 



6X"X" = a positive number. 



Now, the root of the equation X' - 0, of the first degree, is the real 

 root of [1] of the third degree ; and it is equally the root of [3] ; which 

 is also of the first degree, X" being a constant number. Hence, if the 

 roots XX = 0 be diminished by the root of the simple equation X' = 0, 

 that is, if we cause the second term of the proposed equation to disap- 

 pear, the fourth term will vanish also ; they will vanish, moreover, 

 between plus signs ; for X 2 is always plus, and in [2] X n and X are 

 both always plus ; and X" is a positive number. "When, therefore, the 

 imaginaries are equal pairs, the alternate terms, in the transformation 

 which removes the second term, are zeros, each zero being between like 

 signs, and we know that whenever this happens the roots are all ima- 

 ginary. The same has place when the equation is X 2 + iV= 0, N being 

 any positive number, though the pairs are not then equal pairs. 



The conclusion is the same, whatever be the number of equal qua- 

 dratic factors XXX . . . of the above kind ; that is, if the roots of the 

 equation be each diminished by the root of the simple equation X' = 0, 

 the last of the derived equations, the result will be an equation in which 

 the alternate terms will be zeros, each zero being between like signs. For, 

 let the number of equal quadratic factors be n; then, using the notation 

 above, no Xcan appear in any of the derived equations with more than 

 two dashes,because X'" is zero. In the first derived equation, there occurs 

 but one dash in each term ; in the second, there are two dashes in each 

 term ; in the third, three ; and so on up to the final step, in which there 



