Dr. Glaisher on Arithmetical Recurring Formulae. 59 



§ 13. Using the same notation as in § 4 the general 

 theorem may be stated : — 



The numbers given by the formula 



G [d) -(G_i + G n _ 2 ) {d±l) + (G n _ 3 + G,,_ 4 + G»- 6 )(& d± 2) 

 -(G H _ 6 + ... + G, l _ 9 )(d + 1,^ + 3) 



+ (Qtn-io + ... + G»-u)(d,d.±i,d±4:)^&c. 

 all cancel one another with the exception of 



one 1, three 3's, five 5's, . . ., (p — 1) p's, if p be even, 

 a ad 



two (-2)'s, four (-4)'s, six (-6)'s, . . ., (p-l){-(j>-l)\% 

 if p be uneven, 



where ip(p + l) is the triangular number next superior to n. 



§ 14. As an example let n= 9, so that ip(p + l) =10, and 

 therefore p = 4. 



We first form the groups of numbers 











4,6, 



4, 



5, 



4 



2,3,5,9, 2,8 



3, 4, 5, 8, 



3,7, 



3, 4, 6, 



2,4, 



2, 



3, 



2 



1,3,9; (1,2,4,8; 1,7); 



1, 2, 3, 6 ; 



1,5; 



1, 2, 4 ; 



(1,3; 



1, 



2 • 



1) 



0, 1, 3, 7, 0, 6 



-1,0, 1,4, 



-1,3, 



-1,0,2, 



0,2, 



0, 



1, 















-2,0, 



-2, 



-1, 



—2 



in which the central line contains, in the first group, the 

 divisors of 9, in the second group those of 8, 7, in the third 

 group those of the next three numbers 6, 5, 4, and in the fourth 

 group those of the remaining numbers 3, 2, 1. The numbers 

 derived from them, i.e. d + 1 and d — 1 in the second group, 

 d + 2 and d — 2 in the third group, and d + 1, d + 3, and d—1, 

 d — 3 in the fourth group, are then written above and below. 

 The divisors in the second and fourth groups are enclosed in 

 brackets to indicate that they do not belong to the scheme of 

 numbers, being merely written down for the sake of deducing 

 from them the numbers d + 1, &c. Changing the signs of the 

 numbers in the second and fourth groups, we obtain the system 

 of numbers 



-4, -6, -4, -5, -4 

 3, 4, 5, S, 3, 7, 3, 4, 6, 

 -2, -3, -5, -9, -2, -S, -2, -4, -2, -3, -2 



1,3,9, 1,2,3,6, 1,5, 1,2,4, 



0, -1, _3, -7, 0, -6, 0, -2, 0,-1, 



-1,0,1,4, -1,3, -1,0,2, 



2, 0, 2, 1, 2 



all of which, the theorem asserts, cancel each other with the 

 exception of 1, 3, 3, 3. 



