Sir James Cockle on Primary Forms, 351 



while to 2L— M— § corresponds 



(2L-M-|)-L-(i-^ 2 ) + l=-N + i =(E + J) 2 , 



a quantity formed from the bracketed constituents in the last 

 line of art. 26 in the same way that M is formed from L, M, 

 and N. 



33. Take the third lines. To ±-M 2 corresponds M + i, 

 because 



while to 2N— M— -§ corresponds 



(2NlM-|)-(i-^ 2 )-N + l ; 

 = -L + i, =(A-i) 2 . 



34. The roots of the squares are inserted instead of the 

 squares themselves ; and the middle quantities in the system of 

 art. 31 might take the double sign. 



35. From the system of art. 31 a second might be formed 

 by substituting 1— A for A, a third by substituting — (1+E) 

 for E, and a fourth by making both substitutions simultane- 

 ously. With art. 10 the concluding sentence of art. 11 should 

 be incorporated. 



36. Now let us consider cases in which the conditions of 

 arts. 17-20 are satisfied. To this end I symbolize the regular 

 case, that of art. 17, by R; the four cases of art. 18 by K, Q, 

 U, and S respectively ; the first primary form, that of art. 19, 

 by P, and the second, that of art. 20, by p. Then we have 

 seven cases, viz. R, K, Q, S, U, P, and p. 



37. If we arrange these symbols horizontally, and denote 

 the primitive arrangement by 1, the five transformations will 

 enable us to form the following table, viz : — 



l; 



R, 



K, 



Q , 



s , 



u , 



p, 



P> 



/i; 



R, 



K, 



Q , 



U + 2, 



S-2 , 



p, 



P, 



*l; 



P, 



K, 



-(U + l), 



S , 



-(Q+i), 



K, 



P> 



Mi 



P, 



K, 



-(U + l), 



i-Q, 



S-2 , 



R, 



P, 



*/i; 



P, 



K, 



1-8, , 



U + 2, 



-(Q+i), 



P, 



R, 



w-> 



P, 



K, 



1-8, , 



i-Q, 



u , 



P, 



R, 



38. This table discloses the convertibility of regular and 

 primary forms, and leads to another result on which I shall 

 not now dwell. 



2 Sandringham Gardens, Ealing, 

 April 13, 1880. 



