244 THE REV. ROBERT HARLEY ON A CERTAIN CLASS 



which is of the same form as (II) . In this case 



or 



D=-(7^'+I)D^ 



But simple as these transformations are, they do not 

 enable us to pass, at least directly, from the form (A) to 

 the form (C). The first (i°) leads to 



y y 



which is non-linear. And the second (2°) leads to 



which involves an anomaly. These results will, I think, 

 be considered curious and interesting. At all events, I 

 have thought it worth while to record them here, and I 

 shall probably discuss them at some future time. 



14. Every differential resolvent may be regarded under 

 two distinct aspects. It may be considered either, first, 

 as giving in its complete integration the solution of the 

 algebraic equation from which it has been derived, or, 

 secondly, as itself solvable by means of that equation. In 

 fact the two equations, the algebraic and the differential, 

 are coresolvents. In the first aspect I have considered the 

 differential equation (A) in a paper entitled " On the 

 Theory of the Transcendental Solution of Algebraic Equa- 

 tions," just published in the ' Quarterly Journal of Pure 

 and Applied Mathematics,^ No. 20. I have shown in that 

 paper that every differential resolvent is satisfied, not only 

 by each of the roots, but also by each of the constituents 

 of the roots of the algebraic equation to which it belongs ; 

 and that these constituents are in fact the particular in- 

 tegrals of the resolvent equation. In the second aspect. 



