382 



p being of permanent value for the same series, whatever be m } or the 

 number denoting the place in the numerical scale, of any single term 

 [\4] in that series ; thus, for the first series, p = 1 ; for the second, p = 2 ; 

 for the third, p = 3 ; and so on. 



If we take the two terms B, C, next following this m th term [A~\, 

 the first of these, B, will evidently be the expression [A] with the 

 leading factor, m, suppressed, and the new factor {m + p) annexed ; and 

 the next term, C, will be what becomes when the two factors 

 m(m + 1) are removed, and the two, (m +p) [m +p + 1), introduced. 



Hence we have the conditions 



(m+l)C , mB 



B =W+p + h "J =-m+p ... [I] 



.-. [(m + l)AC, mB 2 } = [m+p+l y m+p) AB . . . [2], 



m being the numerical place of A in the series, and p denoting the place, 

 or order of the series itself. 



["We may here notice, in passing, as an inference from the relation 

 [l], that the place (p) of any one of the series being given, we can 

 readily write the entire series from the beginning, or can extend it, 



when leading terms of it are already written, since B = ( m + P^ . an( j 



m 



p, it will be observed, is always equal to the second term minus l : 

 thus, for the third series the first two terms are 1, 4 ; the next term is 



<!±!>? = 10 ; the next (±±312 = 2 0 ; and so on.] 

 2 3 



From the numerical equivalence [2], the proposition enunciated 

 above may be deduced as follows : — 



Let Xp (taken as primitive), and its derivees, be denoted by (X p ) 0t 

 (X p ) 1} {X p ) 2 , &c. : then calling the highest exponent of x in X p} n', we 

 shall have the several conditions (p. 367) for these new functions, by 

 substituting them for the functions, X 0 , X l9 X 2 , &c, in the formulse re- 

 ferred to, provided we put n' for n throughout. But since, by the pro- 

 perty adverted to at the outset (X p ) 0 , (X p ) u &c, are no other than X p , 

 Xp+i, &c, multiplied, in order, by the figurative numberSjl, 1 +p, &c, 

 if A be the m th number in the series, B, C, being the two numbers im- 

 mediately following, then, as (Xp)^ is the m th function in the series 

 (X p ) 0 , (X p ) u (X p ) 2 , &c, we shall have 



A CXp+m.iXp+m+i = (X p ) m _ x (-2^)^+1, and B i X z pJrfn — (X p ) 2 m . 



Now, by the formula [4] at p. 344, the condition of imaginarity for 

 the three functions here last written is 



{m + l)(n' - m~^l)(X p ) m _,{X p ) m ^ > m{n> - m){X p )\ ... [3] 



and the condition of imaginarity for the three functions of which these 



