K-PARTITIONS OF N. 135 



the second above it, in the circle A© ^i ^2 ^3 *4 *5 ^o ^i 



whichisO,— .1,— 4,+3,— 4,— 1,0,— 1 



The greatest value of the constant in 4?^ is when c=2=i, 

 p=:b, and x=i2h-\-6 ; then it is 



_6— 6 + 12(— 1+— 1)=— 36, 



and it is easy to reduce the value of 4P, to the form 



4Px=j^{ar»+3a;*-9x.*2x_i+2ap.*12,_p |, 



where ao=Oy ai=5, at— — 20, 03= — 27, a^=32, 0^= — 11, 



flrg= — 36, 07=5, 08=16,^9= — 27, aio= — 4, an= — H. 

 Thus it is evident that 4P^ is always the nearest integer to 

 the sum of its terms in x. 



We proceed next to find gPx, which will have, besides the 

 portion (sP*) obtained by summing ordinary algebraical terms 

 in equation (C), a second portion (sPx)', which has circulating 

 co-efficients of x, or of Xy or, &c. ; besides that which arises 

 from the summation of the constant circulators in a^x-u a^x-69 

 &c. We begin with 



(5Px)-(5Px-5) = [4Px-l] H. 



5Px= Ax*+ Ba^-\-Ca^ + Dx+ E 



— 5P(x-5)=— Aa?*+20Aa;^— 6-52Aa?H20Aa7— 5^A 



— Baj^+lS Bar— 75Ba?+53B 



— Cx2 + lOCr— 52 c 



— D + 6D 



— E 



Whose sum is (^--:iOH3(._::,l)^ 



_^_3;^+_2^ 

 ■"122 122 122. 



w Uf a __J p_ to r- ^^ n 22 



we ODiam a -go^^' ^""20- 12^' ^■"20^:22' ^"~"20T2a 



Wherefore 



(.Px) =2o:^[i^ + I O.T^ + 1 9.T= + 22^} \- E. 



