1 36 Sir James Cockle on Primary Forms, 



where R(a?) is rational, and therefore of the form g/fcto*^ where 

 (f>(x) is rational. 



4. The second case of the second description, viz. 



\A(D + t3) + {A-2n)(D + u)x*\ 1^1? y=0, . . (5) 



[A] 2 

 reduces to 



g[a]; 



y=o, 



or 



G[A]Jy=0; (6) 



which last is solved, with redundant constants (conf. op. cit. p. 

 421, et Suppl. p. 189), by 



y=x™(C + C,x*+..+C 2n _ 2 x*"-*), . . (7) 



if we make A = D — m. And the redundancy is got rid of if in 

 (5) we substitute for y the dexter of (7). For A and A — 2n 

 respectively annul the terms C x m and C 2n _ 2 # w+2w , leaving all the 

 n quantities C , C 2 , . . , C 2n _ 2 to satisfy the remaining n — \ ho- 

 mogeneous conditions. Thus (5) has a particular integral, which 

 is of the form R(#), and therefore of the form e/W** ; R(a?) 

 being rational and entire, and <\>{x) rational. 



5. The first primary form of Boole {pp. cit. p. 428) may, with- 

 out loss of real generality, be written 



(1 + ^8+^-V^ (8) 



{D(D-l) + [(D-2) 2 -/i 2 >^=0. . . (9) 



But this form is not truly primary when n v an integer or the 

 half of an integer, the latter case corresponding to a quadratic 

 resolvent. And this accords with what prtfedes. The complete 

 integral of (8) or (9) is 



Take n a positive integer and C +1 = C-i; Then y is rational 

 and entire. Take n the half of a positive integer. Then 



n ax 



andifC +1 = l and C_, = -l, th* ^±1 - ^ will become 



n y dx 



teW^+rr+(*- v^r+3(-i)" 



(ff+v^+ir-f-v^+D 



2?i 



