383 



are the multiples (A, B, C), is the condition within the brackets in the 

 following expression, namely, 



( (p + m + 1 )(n - p -~^Tl )X p+m- > (p + m)(n -p- m)X^ m }AB [4] 



and by the property [2] these two expressions [3], [4], are equal ; for, 

 puttig in the. former n - p for n , which it is, we see that the two 

 members of [3] and the two of [4] are respectively what the following, 

 namely, 



(w + I) AC, mB\ 

 [m +p + 1, m + p\AB 



become, when the first member of each is multiplied by the same num- 

 ber, (n - p) - (m - 1) ; and the second member of each by the same 

 number, n - p - m. Hence if the condition of imaginarity hold or fail, 

 for any value of x, for the three functions in [3], the condition will, 

 in like manner, hold or fail for the same value of x, for the three func- 

 tions of the same degree in [4], and, moreover, the two members of the 

 condition [3] are the two members of the condition [4], each multiplied 

 by AB. 



As a practical illustration of this, in a particular case, let X 3 be the 

 function taken as primitive*; then^? = 3, and n' - n - 3 ; also, for this 

 value of p, the figurate multipliers are 1, 4, 10, 20, 35, &c. 



Taking the first triad of these for A, B, C; then the second triad ; 

 and so on, we have 



1st. (20 - 3)10X 8 X 6 > 0 -*4)16X\) = 



{2(n - 3)(X t ) 0 (X 3 ) 2 > (n - 4)(X,)S] = 



{5(n - 3)X 3 X 6 > 4(n - 4)X 4 ')4. 

 2nd. j3(n - 4)80X 4 X 6 > 2(n - 5)100X 5 *j - 



{S(n - 4)(X 3 ) l (X,) 8 > 2(> - 5)(X,V| = 



{60 - 4)X 4 X 6 > 50 " 5)X 6 2 )40. 

 3rd. {40 - 5)350X 3 X 7 > 30 - 6)400X 8 2 J = 



{40 - 5)(X 3 ) 3 (X,) 4 > 3>- 6)(X,V1 - 



(70 " 5)X 5 X 7 > 60 - 6)X 8 , J200. 



and so on, conformably to the general conclusion above, namely, that 

 [3] = [4]- . 



The object of the foregoing investigation is to prove that when a 

 limiting equation X p = 0, derived in the ordinary way from the primi- 

 tive equation X 0 = 0, has imaginary roots indicated between assigned 

 limits, in contracting these limits by Horner's process — whether we ope- 

 rate upon the equation X 0 = 0 itself, or upon the derived equation of 

 inferior degree, X p = 0, the indications of imaginary roots will present 

 themselves, in both operations, at precisely the same Btep of the two 

 processes. Since the numbers, resulting from a transformation in the 

 one operation are all different (except those under X p , and such as 



It. I A. PEOC. — VOL. X. 3 F 



