Sir James Cockle on Primary Forms, 135 



Let 



A(A-r) . .(A— nr)= [A]; +l . 

 Then 



[A]?+i = (A- W r)[A]» = A[A-rp 



so that if in (1) we replace y by its equivalent 



[A]-[y+a»]; 



[a]?[v+2»]; 



then (1) will take the form 



[A-2]»[A + 2 re ]»{A(A-l)- V (V-l)^Y=0, 

 where 



Y= 1 



[A]-[v + 2?i]£ y ' 



Hence the solution of (1) is reduced to that of 



'(A-l±v^)(ATv^)Y»0 (2) 



But a particular integral of (2) is 



Y^-^+l)^"- 1 . 

 Consequently (1) has a particular integral of the form 



(*+l)*E{*)=y, 



where R(a?) is rational. 



3. This form is included in e^ x)dx , where </>(.r) is rational. 

 So, too, is ihat of the solution of the first case of the second de- 

 scription of regular forms, viz. 



{(fc-2»)(D+£)-f A(D + «)ff 8 }y=0, . . (3) 



[A> 

 where A = D+ an) constant. For if we replace y by ^-^Jft 



L^J2 



then (3) reduces to 



[A? +1 G^-„y=0, (4) 



where 



Hence a particular integal of (3) will be given by 

 2/ =[A»G- 1 0. 



But )3-«-2 



Consequently (3) will have a articular integral of the form 

 y=(ff* + l;R(*), 



