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XLIV. Supplementary Paper on Primary Forms. By Sir 

 James Cockle, M.A., F.R.S., F.R.A.S., F.C.P.S., Mem. 

 Lond. Math. Soc, Corr. Mem. Lit. and Phil. Soc. Man- 

 chester, Hon. Mem. Roy. Soc. New South Wales*. 



16, TN my original paper (printed in the Number f for 

 J- February 1875) certain forms of binomial biordinals, 

 regarded by Boole as distinct, were shown to be allied. 

 Boole's four constants, a 1? u 2 , B\, /3 2 , were replaced by three, 

 A, M } and E. The systems are connected, through a and e, by 



1— A— a, A—a=/3 1 , j3 2 ; 



1+E-e, -(E + e) = * 1 -2, a 2 -2 ; 



* x + * 2 -l3 1 -l3 2 =2(a-e) + ±=2(m + l); 



wherein A and E are two-valued and M may take either sign. 



17. These relations give 



A-&=±(1-2A), *!-*, = ±(1 + 2E); 

 and when A and E are integers we have the case of art. 2, and 

 Boole's 1st case (op. cit. pp. 430, 431). 



18. For a— /3 the relations give the four expressions 



^±(A + B) + 1; ^-1±(A-E + 1); 

 whereof I denote the first two by K and Q, and the last two 

 by S and U, respectively, reading the upper signs first. When 

 any one of the four is an even integer we have the cases of 

 arts. 3 and 4, and Boole's 2nd case (ib. p. 431), wherein a 

 factor disappears. 



19. When E is an integer and 2M an odd integer we have 

 Boole's 3rd case (ib.), a so-called primary form. 



20. When A is an integer and 2M an odd integer we have 

 Boole's 4th case (ib.), a so-called primary form. 



21. The distinction between primary and other, say regular, 

 forms may (see arts. 13 and 14) be obliterated by transforma- 

 tion. I shall now show that from a given system A, M, E 

 we may derive five others by the following rule, viz. : — Vary 

 at pleasure the order of the symbols A, M, E, subtracting from 

 a symbol half a unit at each direct step made by it, and 

 adding to a symbol half a unit at each retrograde step which 

 it makes. 



22. These transformations can be effected by algebraical 

 changes of the independent variable x, the change of x into 

 yjrx being accompanied by that of the dependent variable y 

 into (>$r f x)*y. Differentiations with respect to x I shall denote 

 by accents. 



* Communicated by the Author. 



t See also the end of the footnote to my paper " On a Differential Cri- 

 ticoid/' in the Number for December 1875. 



