237 
Ordinary Meeting, April loth, 1862. 
E. W. Binney, F.R.S., F.G.S., Vice-President, in the Chair. 
The Rev. Robert Harley, F.RA.S., Corresponding 
Member of the Society, made the following communication 
<c On the Theory of the Transcendental Solution of Algebraic 
Equations.” 
In my first communication to the Society on the Theory 
of the Transcendental Solution of Algebraic Equations (see 
pp. 181-184 of the current volume of the Proceedings), there 
is a statement to the effect that any linear function of the 
roots oi f{y, x) = 0 must satisfy the differential resolvent. 
This statement requires, I find, some slight modification. 
The differential resolvent for the n- ic equation in y may 
he written as below : — 
• • +x„-4+x„- lJ ,+x„=o. 
Now, if T be a linear function of the roots and of the form 
«o + 0 i yi + <722/2 — 
in which a 0 , a i, a.,, • • • a n are arbitrary constants, it is easy 
to show that 
$(Y) = « 0 X„_ 1 + (1 — «i + «2 • • • + fl n) 
the sinister member of which equation vanishes when «o= 0 1 
and X» — 0. So that the true theorem is — Any homogeneous 
linear function of the roots will satisfy the differential 
resolvent provided that such resolvent is also homogeneous. 
But further, whatever may be the form of the resolvent, it 
is satisfied by any of the constituents of the roots. For we 
know by Lagrange’s theory that those constituents are 
severally of the form 
- 2 y + aft + wb /2 f- to "y n 
n 
Pbocbfdings— Lit, & Phil. Society— No, 1 / 5 .— Session 1861 - 62 . 
