148 REV. T. P. KIRKMAN ON THE 



+(*— 7)(*6 x -409860— *6*_ 1 255840+*6_ 2 275460 



+*6 x _^12960+*6. e _ 4 -141060— *6^-121440)} 



+_*_ | 6(c— l) 6 +136(c— l)*+760(o-l) 3 — 5046(e— 1) 



-_1350(( 7 ~l) 2 +(7-15) 2 +(7-29) 2 )*2 ;e 

 — 1350(( 7 — 8) 2 +(7— 22) 2 +( 7 — 36) 2 )*2^ 

 —.5400(7—1 +7— 1 5+7— 29)*2, 

 —5400(7— 8+7— 22+7— 36)*2 ; _ 1 , 

 (taking and squaring only as many of these six remainders as are positive J 



+ {( C -_l)+( c +6)+(c+13)+...to i terms} 



multiplied singly in order by the th , (c-\-\) th t (c~\-2) th ...ofthe co-efficients 



(7926,— 8424,— 1674,+1176,— 1674,— 8424) 

 of which the e<* is the (e+6/ A , 



+(/*-i+/c+e+/*+i3+ ... to (6(n— 1)+0 terms) | J , 

 where f p : 720 2 is the co-efficient of P*60 x _p in 6 P X , and 



In the above expression the portion 



^* = 720 2 { 4<^ + (/*-i+/*+«+/c+i3+ ' ' t0 7 "~~ C terms )} 



may be disregarded. For it is 



&M -^( /-+/^+--+/e. ) + (/c _ 1+/ ^ + . . t0 



^±_=? terms) j; 



that is, it is always the*sum of h terms f c „ l9 f c+6 , &c, begin- 

 ning with one of the 7 f f x f 2 f 3 f A f 5 f 6 , diminished by 

 about h times the mean value of the whole 60 numbers f v . 

 When x' is small, jS* is the sum or difference of two small 

 numbers. When x' is larger, it is seen by inspection of the 

 table of these quantities f v that no considerable number of 

 them can be taken as f e -i+f c+69 &c, that shall not have a 



negative sum, and shall not with +| • 60 - make the nume- 

 rator of (5 X very far less than 259200=-i720 2 . 



