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XXX. On Binomial Biordinals. 

 By Sir James Cockle, M.A., F.R.S., F.B.A.S., 8fc* 



65. npHIS is the conclusion of a set of papers with various 

 JL titles, but the articles of which are numbered conse- 

 cutively. The latest paper of the set appeared in the number 

 for May 1882 (vol. xiii. p. 357). 



66. The biordinal of art. 62 is, by means of 



e-a U 



v = x a (l-\-x 2 ) 2 U = X a X * 11, 



transformed into 



„ , 1 f 1-P , V 2 V^ 2 1 A 



where 



V = IF-4U, V 2 = V + F-eR 



67. Hence, by the double substitution 



a?=t d and t-iv—y, 

 we obtain 



f + 3sy = 0, (19) 



where the grave accents denote differentiations with respect 

 to t, and (ii'T=l+£ 3 ), 



4 2 l\9 )t 2± T T 2 J 



68. Operating on (19) with -^7 + ^ we get (compare art. 16) 



< y VVN + 3V v + 3^ N + 3(s v +3\% = 0. . . . (20) 



69. Take a new variable f connected with y by 



fkdt fpdt to 



e J y = e J & 

 and giving rise to the terordinal 



r + 8pr+8?r+rf=0 (21) 



70. Then, by equating the criticoids of (20) and (21), we 

 get 



g = s+p 2 — \ 2 + p N — X N ? 



r = 3(* v + 2\s +p?) + 2(\ 3 -p 3 ) +/ V -V\ 



71. When (21) is binomial, then p, q, and r may be taken 



* Communicated by the Author. 



