444 Sir James Cockle on a Differential Criticoid. 



results which agree with both the formulae of art. 5. Again, 



6*x n =;{3n 4 -t-n 3 )x n - 2 ; 



and 



3n d + n~ = 



A n-2 Q n 



[ir - 2 



as a reference to Mr. Scott's valuable paper (ibid. vol. viii. p. 21) 

 will show. This result agrees with the first formula of art. 5. 

 Again, 



3 X ra = (15rc 6 . + 10rc 5 + rc 4 )# w - 3 , 



and (compare Mr. Scott, ibid. p. 22) 



i5 % +io» 5 +» 4 =iL-iL, 



which also agrees with the first formula of art. 5. 

 10. Put x = e x \ then 



0V*= (Sn 4 + 2n 3 )e nx =¥e nx , 



2 



and 



M-l 



6 B e nx = (15rc 6 + 20;* 5 + 6n 4 ) e™= P e 



3 



(see Scott, ibid.) : these results agree with those of Mr. Walton. 

 And we may now write 



i 

 ^X w =HX w " 2 x;+(P 1 -H)X ,, - 1 X a . 



2 2 2 



1 remark that 6> 2 X 2 and 3 X 3 , like 0X, vanish identically, the 

 evaluation being made by substituting, for n, the values 2 and 3 

 in the several cases. 

 11. Next let 



5'^wp*-* •■•<■) 



^-; then 



In (1) change the independent variable from / to # by this last 

 formula and divide the result by X n . Writing the quotient in 



