Sir J. Cockle on JVeic Transformations of Ordinals. 45 



44. Using suffixes in conformity with the analogies of 

 art. 17, and determining 1\ 2 and k 2 from (iv) 2 and (v) 2 , we 

 shall hare (art. 13), 



/(D) 2 = (D + a 2 -l){D 2 + 2(a 2 -l)D + 3/;-a 2 (2o 2 -l)f, 



F(D). 2 =(D+e 2 -l){B' + 2(e 2 -l)J)i-Sg 2 -e 2 (2e 2 -l)}, 



and it only remains to determine f 2 and g 2 . 



45. But 



/ 2 = L 2 + a*-a 2 = L-A 2 + A + a2-a 2 , 



< 72 =X 2 + ^_g 2 = X-E 2 + E+^ + g 2 . 



46. Now (art. 21) we have four sets of relations; and for 

 the first (art. 22) we get 



f 2 = 4Jj + a 2 2 —a 2 , g 2 = 4J$ + el-e 2 ; 

 for the second and third sets respectively (arts. 34 and 35) 



/ 3 =4L+^-o 25 g 2 =^+(e 2 -iy, 



and 



f 2 = L + (a 2 -iy, o- 2 =4X + e* 2 -e 2 ; 



and for the fourth (art. 36), 



f 2 =L + (a 2 -rf, g 2 =X + (e 2 -iy. 



47. The quadratic factor of /(D) 2 can be written 



(D + a 2 -l) 2 -(3a*-3a 2 + l-3/ 2 ), 

 which for the first and second sets is 



(D + a 2 -l) 2 -(l-12L), = (D + a 2 -l) 2 -(§I) 2 , 

 while for the third and fourth it is 



(D + a 2 -l) 2 -(i-3L), =(D + a 2 -l) 2 -(|I) 2 ; 



and there are corresponding results for F(D) 2 , e 2 , 13, and J. 



48. The relation characteristic of the first set has (art. 33) 

 been giyen. That for the second is obtained thus: 



i _3L = (A-i) 2 =o J 2 , or &-%=«?! 



whence (arts. 29 to 32) 



and 



49. The corresponding relations for the third and fourth 

 sets are : — 



U±3J = 0or4; U = 0or4. 



