134 REV. T. r. KIRKMAN ON THE 



We have now to sum the circulating constants in the e+ 1 

 terms 



aPx-i+aPar-s+aPx-D"!" • • • +3^0-1 

 which constants are all of the form ^p**6^_„. If x — 1 = 

 6)W+p, it is evident that *Qx~p-\ is the only circulator that 

 can have value. If sPor contains ^j,**6^_p, aP^.i will contain 

 kp' *Qx-p-u J^x-& will contain kp,^- *6^__p_i, gP^^g will contain 

 ^j,_8* *Qx-p-\ 5 ^^ t^6 ^+1 terms to be summed are 



*Qx-p~l (^p + ^p-4+^iJ-7 + ^p-12+ . 0> 



the subindices always decreasing by 4, upon the circle 



^0 ^1 ^2 ^3 ^4 ^5 ^0 ^1 • • • • 

 Now since every term is identical with the term standing at 

 the twelfth place above it, kj, is identical with fc^.^g^ and the 

 three numbers fcp-f A:p_4+/Cp.8will recur. If we put 



H-l=3A4-e, or A= ^~^('"~^^~^ , (e^3), 



?=0 gives us for our sought sum exactly 



and if ivO, we must add to this 



tV(^p+^p_4+ . . to ^ terms). 

 Now when x is even, p is odd, and v.v., whence, referring 

 to the periodic constant in gP^-, we have for the sum of these 

 e+l circulators, 



^(b- ^^^0 { (-I+3_l).2.+(_4-4).*2._. } 



+TV(A:p4-Ap_4 ... to 2 terms). 

 Adding this to {^^x) we obtain, putting *2^=1 — *2^_i, 



4P;=-rw{^+3a^— 9a^.*2^_i— 3.*4x-i— 6.*4;r-2— 3.*4^_3 



— (4(i— l)+c)-(l-9-*2^_i)+12(A^+Ap_4 . . .toaerms) V 



where a;=12/i-f 4(2— l)+c=6m+/>+l ; cZ5, and kp is the 

 co-efficient of *^x-p in aP^, and kp^A the fourth behind it, or 



