50 Letter from the Rev. Dr. Lardner to Peter Barlow, Esq. 
n congeneric formule produce a rational equation of the mth 
order ; thus, ; 
22+ 7% 2° —a = Bb? is of the 2 order. 
at J wt Jf BHO? =jOl° de secces 
6 
aWYate—-Va—x= 0? §...... 
Are the chances of solution equal for each individual congener? 
I leave that question in good hands, and remain, 
Yours, very affectionately, 
Bath, Nov. 12, 1835. W. G. Horner. 
XI. Letter to Peter Barlow, Esq., F.B.S., Sc., Sc., respecting 
some parts of his Reports addressed to the Directors and Pro- 
prietors of the London and Birmingham Railway Company. 
By the Rev. Dionysius Larpner, LL.D., F.R.S., &e. 
Dear Sir, 
ig was not until my return to London within the last few 
days that I had the pleasure of receiving a copy of your 
Second Report addressed to the Directors of the London and 
Birmingham Railway Company. The previous communica- 
tions which had passed led me to anticipate some collision 
of opinion between us, but I confess I did not expect that 
any difference should exist on a question of a nature so ele- 
mentary as that which you have noticed in your Report. In 
page 87 you say: 
‘“‘ If (as was assumed in the Parliamentary Committee on 
the question of the Great Western Railway) as much power 
was gained in the descent as was lost in the ascent, the odds 
would be made all even. But that assumption is altogether 
erroneous both in theory and practice.” 
And again, in page 91, you say, referring to your theory of 
the deflexion of bars: 
“The only doubt, therefore, which can remain is, how far 
I ought to reject as inconsiderable any increase of power on 
the descending side. This point cannot be met experimen- 
tally, and I am therefore obliged to depend here only on de- 
monstration. The case certainly involves no difficulty of con- 
ception to those acquainted with theoretical mechanics; but 
the question having been seen in a different light by a gentle- 
man of considerable scientific eminence, I should have been 
glad to have exhibited the effect experimentally; but as the 
whole turns upon velocity, this is of course impossible.” 
To those who have taken an interest in the question respect- 
