39 
In deducing from (3) its development, the order of the 
symbolical factors is indifferent, but the two particular inte- 
grals of the development are, I think, obtainable by reversing 
in (3) the order of the symbolical factors. The differential 
resolvent of every such trinomial cubic as that discussed by 
Mr. Hawson is soluble by a change of the independent vari- 
able, and belongs moreover to a comparatively simple form of 
equation soluble by such change. 
The theory of Transcendental Solution has led me to the 
following proposition (theorem) : — 
If an irreducible algebraical equation of the degree n have a 
homogeneous linear differential coresolvent of the order m, then 
any root whatever of the algebraical equation can be expressed 
as a linear and homogeneous function of any other m of its 
root . 
The general demonstration would not be much more diffi- 
cult than or very different from the particular demonstration 
of the case m- 2. The converse of this theorem, I believe, 
is true. 
In such case let a and b be the particular integrals of the 
differential resolvent which (since m~ 2) is by hypothesis„of the 
second order only. Let a, |3, and y be any three of the roots 
of the algebraical equation. Then, since among the values 
that can be assigned, by means of the arbitrary constants, to 
the general integral, the roots of the algebraical equation are 
included, we have three such relations as 
Aa + D5 = a* 
B a + E 6-/3, 
Ca + F6 = y, 
