Mr WARBURTON, ON SELF-REPEATING SERIES. 483 



«|-l 

 P 



Change the place of s^tj , and write the preceding equation, 



o + l|-l p-v\-\ o|-l c|-l t?+l I -1 p-(» + l)|-l J>-0|-1 



P . , P . P_ _ 9 x ( (P-0 . (P- 1 ) . (P- 1 ) . (P- 1 ) j 



JD+1I1 "*"••• + jp-Dll + J«|l " * J t U|l + 1 + 1|1 + "• + 1 p-(0+l)|l T 1P"»|1 J' 



To give greater symmetry to the expression, change the factorials with diminishing factors, 

 into their respectively equivalent factorials with increasing factors ; and the expression will 

 become, 



» + l| + l p-»|+l »| + 1 o|+l p-(v + l)\ + l p-v\ + l 



fp-«] ["+!] ,[P + l-«] a \[P- v 1 , , [t> + 1] 



■»-»] _ o y [p-«] . [> + *] . ; l 



i*n \ i"ii •" ip-(»+i)ii + IP-"! 1 /' 



t)| + l 



Now in order that the term , x may coincide with the term of the middle 



»| + i 

 column, g-rj , take (p + 1 - v) = (n + l) ; that is, take p - v = n. Then the ex- 

 pression will become, 



« + l| + l n| + l o| + l o| + l n-l| + l n\X 



n 



j0+i|i + — + 1 ^|i + jcli \i»U + "* i»-i|i l"! 1 /' 



And since p m n + v, it is plain that 



O+ll+l n| + l 



{? + + ^±il_i 



,l«+lll + + l»ll j 



(26) 



n + v 



is the sum of the terms which in (l + t) are omitted, in order to produce Diagram IV. 

 from Diagram III. ; and that 



t)|+l n-l|+l n|l 



in (v + 1) v 1 



|"l»Ti + + jn-iii + pryij 



n + v — 1 



is the sum of the terms which in (1 + 1) are omitted, in order to produce Diagram IV. from 

 Diagram III. » 



n + v n+l — v n+v— 1 re+2— v 



And in Diagram III., (l + t) is the factor of A0 2 "+ 1 ; and (l + t) is the factor of ACr 2n + 1 . 

 Therefore, in Diagram IV., 



o+l|+l »|+1 



r n + v in (t> + l) h 



n + l — v 



is the coefficient of A0 2n + 1 ; and 



o| + l n-l|+l n|l 



~n + v-lt n (p + 1) V IT 



n + 2 



is the coefficient of A0 2ri + 1 . 



