Sir James Cockle on Primary Forms. 137 



In either case* - ^ is rational, and —■ in the former rational 

 y dx ax 



and entire. In both cases, therefore, y is of the form ef^ x)dx y 

 where <f> (x) is rational. The proof may be extended to the cases 

 of n, a negative integer, or the half thereof. In no other cases 

 has (8) or (9) a particular integral of the form e^ {x)dx . And 

 since (8) and (9), if soluble through Boole's reduction, must fall 

 under one of the three forms (1), (3), or (5), the first primary- 

 form is not in general so soluble. The same may be proved for 

 his second primary form. 

 6. Let 



3 + >i + ^=° (">) 



where z is any function of x. Suppose that 2a= — J- -j_ C\/z 



dz 

 where p = -^-. Then (10) is reducible to an equation with con- 

 stant coefficients by a change of the independent variable. The 

 primary (8) is thus soluble. But it may be otherwise solved. 

 Let j^U mean §Vdx, and let P(Jj: § x rj) mean the double 



operation ff f 17 ff f#. . continued in infinitum. Also let y=zyjr(x) 



be a particular integral of (10). Then if f be determined from 



d£ 

 the equation f of the ccesura, viz. -j- H-2af=0, and rj from 



* The same is true of every case. For (1), (3), and (5), together with 

 the three forms 



{(A-2m)(A-l)-aV}y=0, ( a ) 



{(A-2n)(D + /3)-hAz 2 }y=0, (b) 



{A(D+/3) + (A-2rc> 2 }y=0, (c) 



and the six other forms deduced from these by the change of a? into - con- 

 stitute the twelve forms which the reductions of Boole solve. Both here 

 and in the text m and n are integers. The arguments for (a), (b), and (c) re- 

 spectively are analogous to those for (1), (3), and (5), and show that there 

 is at least one particular integral of the form e/^W'fe, 



f This method of synthetical solution may give a finite result, a series 

 summable or otherwise, or a suggestion of the form of ^r{x), in which a 

 constant or constants are to be determined by substitution. For the ter- 

 ordinal * 



.%■»&*»%+-* » 



where a, b, and c maybe variable, we assume y = PJ^J^J^C • ^K#)> where 

 \K#) is a particular integral, say zero, and £ is determined by the casura 



