80 



irig the integral roots of numerical equations upon the following 

 theorem. 



If the equation 



x n +p 1 x" 1 ' 1 +p. 2 x ll ~' 2 + &c. +PiX n ~'+ &.c.+p n =0 

 has a positive and integral root m, we shall have — p n equal to m 

 terms of the following series : — 



A n _! + 1. A B _ 2 + 1.2. A„_ 3 + 1.2.3 A B _ 4 + &c. +1.2. ..(«—«+ 1) A; 

 + &c. +1.2...(»— 1)A 

 + A„_ 1 + 2.A n . 2 +2.3A„_3 + 2.3.4A ) ,_ 4 + &c. +2.3 ...(n-i+2) A s 



+ &c. + 2.3...rcA 

 + A ll _ l + 8.A„_ 2 +3.4 A„_ 3 + 3.4.5 A n _ 4 + &c. +3.4... (m— i+3) A. 



+ &c. +3.4...(» + l)Ao 

 +A„_ 1 + 4.A„_ 2 +4.5A„_3+4,5.6 A„_ 4 + &c. +4.5...(n~i+4) A, 

 + &c. +4.5...(w + 2)A 



&c. &c. 



where 

 A i =(n — i)(p i —h 1 p i _i + h 2 p i _ 2 + 8LC. + (—iyh,.p i _ r + &.c.±h i _ l p l + h i ) 



and 



Aj = the sum of the natural numbers 1, 2, 3 (n—i). 



A 2 =the sum of the homogeneous products of the same quantities of 



two dimensions. 

 A 3 =the sum of the homogeneous products of the same quantities of 



three dimensions, and so forth. 

 From the above formula for A t may be determined all the coeffi- 

 cients A except the first, which is determined from the equation 



A„_ 1 =p„_ 1 — p n - 2 +Pn-3 — &C.±p 1 +l. 



Having determined the quantities A in any particular case, let them 

 be substituted in the first line of the series. If the sum of that line 

 be equal to —p n unity is a root of the equation. Let the second line 

 be then written down and added to the first. If the sum of the two 

 equals — p n , 2 is a root of the equation, and so by adding successive 

 lines we shall ascertain whether the successive integers 3.4., &c. are 

 or are not roots of the equation. 



The quantities h in the expression for A. ; depend upon the number 

 of the coefficient and the number of the dimensions of the equation. 

 The author proposes that these should be calculated and tabulated 

 for equations of all dimensions up to a certain limit, by which means 

 we should be in possession of so many skeletons of equations, ready 

 for application in any particular case, and the calculation in particular 

 instances would be thus greatly facilitated. 



It will be observed that each successive line is derived from that 

 preceding by a simple division and multiplication of the separate 

 terms of the latter, and thus each succeeding trial facilitates those 

 which follow ; contrary to what obtains in the ordinary method by 

 successive substitutions, in which each attempt proceeds de novo. 



If the addition of a term makes the series from being greater than 

 p n less than it, or vice versd, a fractional or surd root will lie between 



