7-PAKTITION8 OF X. 147 



Let now 



The equation D is 



for the constant fi x , depending only on the seven values of c 

 and the circulating constant in P X , can have only 420 

 values, and /3«=/3*-49o, which therefore disappears in D. 

 Wherefore 



as has been already proved, and 



— _ 2 ? 243 

 a ~ 42.720 3 ' 



The constant j3* is given by putting x=.x'> ^420; which 

 requires 



( 7 P,0"=[eP«'-i]"+C«P/-«]"+ • • +[«Po-i] W , U 

 <rf+P*=j7filf>-i+f> + *+ • • • to/.U inclusive), 



A== S353> { 27243 *'+42(/^+A fl + • • to/.U) j J 



This being added to ( 7 Px)+( 7 P*y found before, completes the 

 expression of 7 P* . 



We have then finally, writing 

 x=420m+a?' 

 =420*f>+42(*— l)+ 7 

 =420m+42(»— l)+7(*— l)+c ; 

 (•70, Z7), (c70,Z8), (770, Z43), (*70, Z420), 



7P ^[iTw{ 2( ^-^ +84(+ ^^ )+1120 ^ W)+437 ^ W) 



— -9527(*>— c 8 )— 55899(*—c) } 



""42^ ( 1350(«»-7 3 )+(37947-28350 *2^ 1 )(* , -7*) 



+2724S(*— *') 



