365 



be squared, the first two terms of the result, when multiplied by the 

 proper Newtonian factor (as suggested by the degree n of the equation), 

 will always be the same as the first two terms of the product of the 

 extreme terms when multiplied by the corresponding Newtonian factor. 

 Let any consecutive three of the functions [2] be represented by 



Ax p + A'x p ~ l + . . . . 

 -pAx*- 1 + -(2? - l)^^ 2 + . . . . 



m{m +1) 



- 1) (p- 2)A'xp- z + . 



The first two terms of the product of the first and third of these ex- 

 pressions retaining coefficients only, are — 



m(m + 1) 



or 



p(p - 1)A* + — -(p - \fAA> 



[3] 



and the first two terms of the middle expression squared, coefficients 

 only being retained, are — 



LfA* + -Mp- l) A A' 



[4] 



By multiplying [3] by — — . -, we get [4] : therefore if we 



multiply the former expression by (m + l)p, and the latter by m(p - 1), 

 the results will be the same ; and these two multipliers are the proper 

 Newtonian factors, as is easily seen by putting 1, 2, 3, &c, in succes- 

 sion for m, and n, n - 1, n *- 2, &c, in succession for p. 



The same conclusion may be arrived at more expeditiously thus — 

 If / (x + r) were a power, that power would, of course, be 



(x + r) n = [x + ~ x n + ^jp 1 xn ~ l + • • • • [5] 



and the triads of [1] would all be triads of equality. The square of a 

 middle term, multiplied by its Newtonian factor, would be a result 

 which, in all its terms, would be the same as the product of the extreme 

 terms, multiplied by the proper factor. But though / (x + r) be not 

 equal to (a? + r) n , f (a?), and, therefore, all the functions [2] derived 

 from it will have the first two terms of each the very same as if the 



