358 



Thus— 



When n is even, 



A n x n + 0 + 0 + 0+ ....+0±^4 0 

 + - + - + 



-TV- . , , \A n x n + 0 + 0 + 0+ .. ..+0 + ^o 



When n is odd, \ + _ + _ - 0 r +, indifferently. 



For taking account of the leading sign + in the underwritten row of 



signs, it is readily seen that the n - 1 zeros, when n is even, furnish - 



n 



changes from + to -, if A 0 is +, and - - 1 changes if A 0 is -. And 



n-\ 



that when n is odd, the number of changes is — - — , whether A 0 is + 



or - : and since each change from + to - implies a distinct pair of 

 imaginary roots, the number of such roots, thus indicated, is precisely 

 the same in each case as the number determined above. As the sign 

 under the zero immediately preceding the last zero is always + when n 

 is even, and always minus when n is odd, in the latter case it is plainly 

 matter of indifference which sign be placed under the last zero. 



But suppose that two or more significant terms precede or follow 

 the zeros, or both precede and follow. By taking the successive limit- 

 ing equations, these latter terms will disappear one by one, till only a 

 single significant term beyond the zeros is left ; and by reversing the 

 coefficients of the equation thus reached, and proceeding in like manner, 

 we shall finally arrive at an equation of the form 



A'ntf 1 ' + 0 + 0 + 0f... + 0 + JL' 0 = 0... [2], 



that is, at an equation of the same form as the equation [1] above. 



(22) If, in this latter equation, A' n > should be positive, like A n in 

 [1], the foregoing conclusions, as to the number of imaginary roots, 

 would of course apply to it ; but if A be negative, we should have to 

 change the extreme signs of [2], or to multiply the terms by - 1, 

 before we could deduce the number of imaginary roots, as above, from 

 the underwritten signs. Yet, leaving the leading minus sign and the 

 sign of A'o unchanged, if, as before, we write + under the first term, 

 and - under the first zero ; then + under the next, and so on, as directed 

 above, till we come to the last zero, and write under that + or - , ac- 

 cording as A'q is + or - , it is easy to see that the underwritten row 

 of signs will have the same changes from + to - as if the signs of A' M , and 

 A' 0 had themselves been changed ; for it will be remembered that, in 

 the case of n' being odd the changes from + to - are the same, which- 

 ever of these signs be placed under the last zero. Now, since the 

 signs of A' n < A' 0 , are the same as the signs of the original coefficients 

 from which they have been derived, and as the intervening zeros are 

 the same in number as in the original equation, it is plain that actual 

 derivation is not necessary. All we have to do is to proceed with the 



