K-PABTITIONS OF >'. 139 



where ar=5.(l2A+fc—l)+c=12r-fp+l; c70, A 7O ; and 

 dp^ «p*7» «j>»8» &c., are the ;)^ (p+7y*, (p+H)^ &c., of the 



circle Oo «i «« * ' ' «io «ii «o ' * * > the co-efficients of j2« 



(l2x_o, 12x-.i, &c.), in 4P*; that is, Op, 0,^7, &c., are, in that 

 order, the series, 



Oq O/j Qi Qq 04 an a^ a\ a^ as Oio as 

 5 —20 —27 32 —11 36 5 16 —27 —4 —11, 

 of which k terms are to be added together, beginning with «p, 

 and then multiplied by 20. 



There is no difficulty in reducing the terms free from x, the 

 periodic constant, to the form Sc(p.*60j,_p, where 



do = 0, d^ =-|- d,d^ zz+104, (/g =—351,^4 =—6lO,d^ = 905, 

 <f« =— 216, (f, =— 361,^8 =— 256, d, =+ 9, (/,o=+3t30, rf^^zr— 31, 

 rf,a=— 676, </,.=+ 9,(/,^=+104,(/,^=+225,<f,8=— 576, J,,=-|-329, 

 dj8=_216, rf,9=— 351, <fgo=+320, </2i=+ 9,(/22=— 216, </2a=— 31, 

 </j^=_576, rfa4=+585, (/26=+10'i. </27=— 3^1' <^28=— S76» <^29=+329, 

 rf3„=+300, (/, ,=-351, c/,2=— 256, c/8»=+ 9, ^/a^zr— 216, (/35=+545, 

 rf.8=-576,</3,=+ 9, (/38=+104, (/3„=_35I,rf^o=+ 0,e/^,=+329, 

 </^a=— 216, rf^3=— 351, d^^z=—256, d^^ = -{-585, d^^=—2l6, d^,=— 31, 

 </^3=_576, (/^8=+ 9, rf,„=+680, rf,,=_361,(/52=-576,<f,3=+329, 

 </«♦=— 216, </5.=+225, </,3— —256, rf^,=+ 9, rf^8=_216, rf„=— 31. 



These numbers are obtained from the formula thus ; e,g., 



x=60A+32=5-(l2^+7— l)+2=12r+7+l 

 ar=60A-h40=5(12^+8— l)+5=12r+3+l 

 a;=60A+49=5(12^+10— l)+4=12r+04-I. 

 The first has A= 7, /?=7, and *5^_2=l. 

 The second has A= 8, p=3, and *5^ =1, 

 The third has k=z\ 0, p=0, and *5^_i= 1 =*2^._, ; 

 therefore cr3s=6-1304-4+20{(a7-|-a2+O9+a4+«ii+a6+ai=) 

 —52} =784— 1040=— 256 



d4o=7' 130 + 1 30—20 { («3+<7io+«5+ao+a7+a2+a9+ 



04=)— 52} =1040—1040=0 



(/49=9*130 + 135— 36+20{flfo+a7 + «2 + flr9-|-a4-i-a,i 

 +aS+a,+flr8+a3=)— 63} =1269-1260=9. 



