345 



putting 2 m for n, the middle term will be the mth, and therefore, by 

 the general form referred to, the proper multipliers will be 



(m + 1) (2m - m - 1) = (m + l) 2 , and m(2m - m) - m 2 . 



Of course, since the multipliers in each completed series taken in 

 order from first to last are the same as when taken in order from last 

 to first, the product of the extremes, as also of any two multipliers 

 equidistant from the extremes, will be a square number. 



(6) If in any one of the foregoing conditions, the first member 

 should be equal to, instead of greater than, the second, a pair of ima- 

 ginary roots will still be implied. For in that case, not only does the 

 second coefficient vanish in the equation in which the condition 

 of equality has place, but the third coefiicient also — the general ex- 

 pression above, for the third coefiicient, being then zero ; and we know 

 that consecutive zeros always imply imaginary roots. 



(7) Any one of the group of conditions [3], involving three conse- 

 cutive coefficients of the primitive equation [/], being taken, if that 

 condition hold or fail, it will, in like manner, hold or fail for the terms 

 involving the same three coefficients in every limiting equation derived 

 from the primitive. This is proved as follows: — 



The general expression for any triad of consecutive terms in [1] is 



The triad derived from this is 



(m+ 1) A m+l x m + m A m x m ~ l + (m - 1) A m . l x m -' i ) 



and the property affirmed is—that according as the condition in [4] 

 holds or fails for the above primitive triad, so will the following condi- 

 tion, in which the derived triad replaces the former, hold or fail, 

 namely, the condition 



m(n -m-2) (m - 1) (m + I) A m . x A m+l > (m - 1) (n - m - 1) (mA m )*, 

 or, expunging the factors common to both sides, the condition 



(n - m - 2) (m + 1) A m _ x A m+X > m(n -m- l)A 2 m . 

 For, it being remembered that, for the triad with which we are now 

 dealing, the degree of the equation is n - 1, and not n (as in the case of 

 the primitive triad), we must put n - 1 for n in this expression : its 

 form will then be 



(n - 1 - m - 2) (m + 1) A m _ x A m + x > min - 1 -m - 1) A 2 m , 



which is the same as 



(m + 1) 0 - m - 1) A m ^A m+l > m(n - m) A\, 



that is, it is identical with the form [4]. And, as a corollary to this 

 theorem, it follows that when the two expressions [3] are equal instead 

 of unequal for any triad in a derived equation, they must be equal for 



