L 44 ] 



V. New Transformations of Ordinals. By Sir James Cockle, 

 M.A., F.R.S., F.R.A.S., $c* 



37. TT follows from my paper in the Number for last Sep- 

 J- tember (p. 189) that, when certain conditions are 

 fulfilled, we get new transformations. 



38. Before showing this, I give a brief verification! of some 

 of the calculations. 



39. We have identically 



N 2 -L 2 =N-L + a>(n-l); .... (9) 



and the values of B 2 and C 2 + b 2 are given in art. 20. 



40. Art. 17 yields the system 



C 2 + b 2 Q B 2 



which the relation 



b + 3a>(co + l) = (10) 



reduces to 



-(2a> + 3) = 77 2 =-(2a> + 3). 



41. But, by (3) 2 and (10), 



V 2 = -i {3 + (4co + 3)}. = -(2o> + 3) or 2a. 



42. If, therefore, we take the radical positively, all the con- 

 ditions of art. 17 are fulfilled, and we determine oo from (10), 

 thus (see art. 32) obtaining : — 



U , U 



co=--lov - j; 



U IT . 



%= ___lor ¥ -3. 



43. The other two forms for r) 2 , viz. — — 2 and —-5-, are 



improper, and in general irrelevant. If, however, co =— f, 

 one of the improper forms may be relevant, and only one. 

 Both cannot be so, because, when w takes a single value, that 

 value is —%. 



* Communicated by the Author. 



t To verify art. 10, let b = 0, and take the radical negatively. Then 

 (1), (2), become B, C = 0, and (3) is an identity. But B, C = together 

 make x 3 X.t linear in x n , while 6 = makes x 2 Xs linear in X* ■ or, in" other 

 words, the terordinal in z is binomial as it stands. I have verified arts. 19 

 and 20 by the substitutions 



A,E = 0,0; 1,1: -1,-1; 2,1. 



