350 Sir James Cockle on Primary Forms. 



27. Since our transformations operate differently on x and 

 on y. we may, i|r being any one of them, put 



The transformations / and <£, so far as they affect x only, are 

 of the second order ; and although fcpx and cpfx are of the 

 third order, yet (f<f>) 2 x = (j>fx and ((fcf) 2 x=f(j)x. Moreover 



28. Since 



/ changes x 2 into x~ 



$ 



)) 



a? „ -{! + «?), 



ft 



n 



a? „ -(1+^- 2 ), 



¥' 



» 



a? „ -(l + O- 1 



fV 



» 



x 2 „ -(1+^- 2 )- 



we may pass directly from the first of these forms to any other 

 by making the appropriate changes of variables and then 

 taking the criticoid. And any one of these forms indifferently 

 may be made the first form. 



29. It remains to be indicated what changes in A ; iE, and 

 E correspond with those made in L, M, N. 



30. Since the system 



A 2 -A=-L, E 2 + E = -N, 



is, save in form, the same as 



(A-1) 2 + (A-1)=-L, (E + 1) 2 -(E + 1)=-N, 



an interchange which sends N to the extreme left and L to 

 the extreme right will not disarrange our formulae, provided 

 that we replace A and E by E + l and A — 1 respectively. 



31. In order to see that the following system, viz. 



/V, (B + l, M, A-l), 



</>V, (M+i, A-|, B), 



,#V, (B + l, A-!, M-i), 



*/V, (JB+i, B+l, A-l). 



ftfY, (A, E + i, iE-i), 



corresponds, line for line, with that of art, 26, it will be suffi- 

 cient to point out how two of these lines are constructed from 

 the corresponding line of art. 26. 



32. Take the last lines. To L corresponds A. To J— M 2 

 corresponds M — J, because 



(iE-£)(,E + i)=-a-^ 2 ), 



