298 MR. W. H. L. RUSSELL ON THE 



I now proceed to the solution of the general equation, 

 where 



^^^~ D(D-i) (D-2) . . . (D-7Z + 2) 



The solution of this equation may he expressed in series thus; 



y=Co{Ao(°> + A/°V-^ + A/°V(^-^)+ . . . } 

 + C,.«'{A/^) + A/^)^^-» + A/»V(^-0+ . . . } 

 + C^a?HAo<^> + A,<*)a?'^-» + A^(^V('^-0+ . . . } 

 + &c. 

 + C^»-{Ao(*'> + A/'*).x?^-'+ . . . A^(')a?"^(»*-0+ . . . }, 



where A^W = 



(Zn—l\/ 2%— l\ / 2W— i\ 



(^r4-(M— i)) fr-\-z{n — i)^ (r-\-m{n—j)) 



(^n—z\ / 3W — 2\ / qw — 2\ 



(r-j-(w— 2)j (r-\-{n—2)-\-{n— i)) • . . r?'+(w — 2)+(w— i)(w— 1) j 



(r+(n- 1)-^^ ('r+2(^- i)-^^ • • • (r+m{n-j)-^^\ 

 ^r+(w-3))(r-)-(w-3)+(%-i))...(r+(w-3)+(w-i)(w_i)) 



r-j-(^_i)_ \lr-{-z(n-i)- \lr-^m{n-i)- \ 



(r+i)((r+i) -!-(%- 1)) . . . . (»'-f.i+(m- 1) (w- i)) 



This term can be transformed by a method similar to 

 that which we employed for the quartic resolvent, and we 

 find this expression equivalent to 



Ur-^i) . . . r-\-m{n— i)iUr-^m(n—i) — n\(r-^('m—i)(n—i)—n\. .. (r-\-n — z)\ 



^ f (r—i)n 1 



^ ^''"' * r r— I 1 „ r ^ * 



rJMH IT )r-\-m{n—i)-j-i [ 



