Sir James Cockle on Primary Forms* 139 



expressions which give M — L — N the value 



(k—h) (f—g-\-a — e)+n(ka—he) + (a— e)*. 



9. If h and k are finite, then by changing x into (-r) n ff, divi- 

 ding, changing x into x n , and making appropriate changes in the 

 constants, we may without loss of real generality replace the bi- 

 nomial by 



(l + *>*0 + 2(a + ^V<| + (/+^% = O, . (11) 

 the trinomial by 



(l + tf 2 )V^f+(L + M^ + N*%=0, \ . . . (12) 



and the accompanying system by 



L=f+a-a 2 , (13) 



N^ + e-e 2 , (14) 



,M=f+g + 3a — e — 2ae, .... (15) 

 M-L-N + l = («-e-H) 2 =JE 2 . . (16) 



10. Boole's process reduces (11) to 

 {D(D-~l)+2aJ)+f}^ 



+ {(D-2)(D-3)+2*(D-2)+^}A=0, . (17) 



and if we put 



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



then (11) is solved through Boole's reductions : 



First, if A and E are both integers ; 

 Secondly, if M — A— E is an odd integer. 



And (11) is primary : 



First, if A is an integer and 2JE an odd integer ; 

 Secondly, if E is an integer and 2JE an odd integer. 



11. In (12) change the independent variable from x to tana? 

 and take the criticoid of the result. Then (12) is replaced by 



+ (Lcot 2 #+M+l+Ntan 2 %=O. . . (18) 



Now, if by a factorial substitution we pass from a form in which 

 the last coefficient is variable to one in which such coefficient is 

 constant, the two forms may be called conjugate. A conjugate 

 of (18) is 



0+2(Acot* + Etan*) J^ + Qy=0, . . (19) 



