Sir James Cockle on a Differential Criticoid. 443 



Operate on either side with X -7-, and the dexter becomes 



(XX 



n—r+1, 



But 



/ d \ r] n - r+1 ?/ 



s(x«"+x s rT)y;. 



^.^-X^V^X"; 



and such dexter may be written, omitting the last factor, 



SX(l + Y)^X ra = S£ r X* +1 , 

 and therefore 



«-H jn—r + l 



X^ 2/ = S^X w+1 ^ J. 



dx) J dx n - r+l 



Thus, if the development holds for n, it holds for n-\-\ and is 

 generally true. It holds for ?i = I ; and I remark that 



6X n = ( jnX n - 1 X l = n q K n - 1 X l , 



which vanishes when ?z=l, explains the suppression of $ M X W . 



7. Proceeding, we find 



6*X n = an 2 {n- l)X n " 2 X?4- crn q V l - l X q ; 

 and since 



crn^n — l) =3n 4 + n 3 , 



consequently 



^X"=(3« 4 + W3 )X B - 2 X?+ % X"- , X a . 



8. Proceeding further, we have 

 6 3 X. n = <T[{3n 4 + r l3 ){n-2)X n - 3 X 3 ] 



+ j2(3n 4 +« 3 )+(n-i) % }r- 2 x 1 x 2 + % r- 1 x 3 ]. 



Now 



<T(3n 4 + n 3 )(n — 2)=o-(lon 5 + 6n 4 + 4n 4 + n 3 )=I5n 6 +l0n 5 -\-7i 4 . 

 Again, 



cr(6w 4 + n 3 {n + 1) ) = a(lQn 4 + 4w 3 ) = 10n 6 + 4ra 4 . 

 Hence we have 



<9 3 X n = (15» 6 + 10» 5 + 7i 4 )X n " 3 X? + (10n 5 + 4K 4 )X n - 2 X 1 X 2 



+ « 4 x n - 1 x 3 . 



9. Put X = <27; then 

 and 



