532 Mr. J. Cockle on the Correlations of Analysis. 



by equating it with 



so that, replacing </>(a ; a) by a, we have 



{£—>-♦<•■ a « 



and so on. Then 



£(**«jy ••**;?) = ° (4) 



is satisfied by 



and, if a, a 2 , . . ct n be w particular integrals of a linear differential 

 equation of the nth. order, the development of (4) will give a 

 symbolical decomposition of the differential equation analogous 

 to the decomposition of an algebraical equation into linear 

 factors. 



The symbolical decompositions of a linear differential equa- 

 tion are, in general, infinite in number. If it denote the " sym- 

 metric product/' and 6 the " resolvent product/' there are 

 decompositions of the differential resolvents of cubics, biqua- 

 dratics, and quintics which may respectively be represented by 



£(A*,BAgW)=0, 



a, /3, 7 being constituents of the roots, and A, E, C, D arbitrary 

 constants. 



2. There are critical functions of the coefficients of differen- 

 tial equations analogous to those "critical functions" of the 

 theory of algebraic equations which, in the theory of quantics, 

 are termed leading coefficients of covariants, peninvariants, or 

 seminvariants. Let uv be substituted for y in the linear differ- 

 ential equation 



(1, bjc/d,.,. J^, l) B y=0, ....(«) 



