46 Sir J. Cockle on New Transformations of Ordinals. 



50. We may state all our results concisely as follows : — 

 When I, U, J are connected by one of the relations 



U-3(l-0 2 —)I + 3(l-0 2 - re )J = 4, . . (11) 

 U + 3(l-0 2 — )I-3(l-0 2 -»)J=0, . . (12) 

 the terordinal in z can be transformed into 



(d+, 2 -h-Aj) ( d+, 2 -i)(d+, 2 -i-^ ) _ 



(B + a 2 -l+~jyB + a 2 -l)(B + a 2 -l-^ n l) 



.... (13) 

 wherein, for the first set m=l = n; for the second, m=l, 

 n = 2 ; for the third, m — 2, n— 1; and for the fourth, m = 2 = n. 



51. Both I and J are independently bisignal; and, without 

 affecting I or J, we can change U into V ±4k, where k is an 

 integer. The restriction in art. 35 disappears. 



52. When any one of the conditions implied in (11) or (12) 

 is satisfied, Boole's algorithm would enable us at once to form 

 a terordinal having its factors, as well in the numerator as in 

 the denominator, in arithmetical progression. But (13) yields 

 as many results again ; for, since a 2 — e 2 has two values, (13) 

 has two forms. 



53. Take U = 0. Here y 2 = — 1 or — 3. The result yielded 

 by —3 may be always transformed* into that obtained by 

 Boole's algorithm ; not so that yielded by —1, which is 

 peculiar to the criticoidal transformation. 



2 Sandringham Gardens, Ealing near 

 London W., December 15, 1881. 



Errata in Vol. XII. (additional). 



Page 189, art. 2, line 2, first term, for d 2 read d 3 

 „ 190, last line of text, for b read —b 

 „ 195, line 2, for -or read m 

 „ 196, art. 34, line 3, for L read 3L 

 „ „ „ „ 5, for N read 3N 

 „ „ art. 36, line 3, for -2 read or 4 



* This transformation is direct, viz. effected by Boole's process. The 

 criticoidal transformation is indirect ; -viz. we connect two terordinals by- 

 means of a biordinal. I have applied Boole's algorithm in the ' Messenger 

 of Mathematics ' for November 1881 (pp. 109, 110). 



