237 



Ordinary Meeting, April loth, 1863. 

 E. W. BiNNEY, F.R.S., F.G.S., Vice-President, in the Chair. 



The Rev. Robert Hahley, F.R.A.S., Correspondini? 

 Member of the Society, made the following communication 

 ^^ On the Theory of the Transcendental Solution of Algebraic 

 Equations." 



In my iirst 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) — must satisfy the differential resolvent. 

 This statement requires, I find, some slight modification. 



The differential resolvent for the 71-\g equation in y may 

 be written as below : — 



*(.)=£a+x.£i...+x„.2+x,._„+x,.=o. 



Now, if Y be a linear function of the roots and of the form 



in which a^y «i, cio, •••«,» are arbitrary constants, it is easy 

 to show that 



$(Y) = rtoX,_i + (l— ai-i-«2-..+«„)X„ 



the sinister member of which equation vanishes when «o=0 

 and X„ = 0. So that the true theorem is — Any 1iomoge7ieons 

 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 



PnocBEDiKcs— Lit. & Phil. Society— No. 15.— Session 18G1-62. 



