1 38 REV. T. P. KIRKMAN ON THE 



next twelve functions, so that I2h successive functions i^x-u 

 &c., will contain h times the term 



h times for every value of j9. 



If then c+1 be 12A+A-, x=ifie+c, giving 



,=J_(._5(/._l)_c), 

 the sought sum of the constant circulators will be 



j-22g^.(^— 5(/c— I) + c^S«r fromr=0 to r=ll, 

 + ~ Sa,(*12^_,.^+ • • • • +*12^_i_5(,-i)-,), to & terms 



that is, if 0;= I27W4-J0+1, sinceSarr=ao + «i+ •** +^n= — '78j 



2Qx 26(5(fc^l) + c) 

 12-220"^ 12.220 



+ j2-^(«i)+«2>t7 + «P^2«p+9+j to k terms); 



for *l2^_p_i, the only circulator that does not vanish, is 

 affected by ap-^Up^-j, &c., in that order, the subindices increasing 

 by 7, or, which is the same thing, retreating by 5, upon the 

 circle (qo «i «2 * * «io »ii c'o * ' )• Adding now the terms just 

 found to (5P^) + (5P^)', putting 90^— 90.r.*2^ for 90*2^_i 

 found in (sPa?)'? and remarking that, in the last found ex- 

 pression, 



26c=26.*5^_i+52.*5^_2+78.*5,:.3+104.*5;.-4+130.*5^., 

 we have finally, 



5P.=2^-J2;4+I0:r3+10.^— 30^— 90.r*2, 

 +135.*2;,-i+130(/c— 1) 

 +130.*5;^126.*5^.i+4*5^.2— 86.*5,_3— 36.*5^>4 

 +20[ap+ap+7+«pt2+ * • * +«i)47-(fc-i) ; (to h terms)]!; 



