54 Mr. Davies on Mr. Herapath’s Demonstration. 
this admission? Does not P. Q. perceive, so long as 7 yaries 
by integer values only, that v varies through a system of fractions 
whose common difference is an integer, and is altogether inca- 
pable of any other system of values whatever ? 
Does it need to be urged 
whilst ; Av = An, and 
: i aoe ae 
An = integer : that Av is also = integer? 
Take, for “ example,” r = 4, and n = integer: then v = 2 
— 1; and so long as 2 retains its integral character, v can never 
become 7! + 2, nor 2"! +/—1. The only system of values 
which it admits, is comprised in the expression m + + [7, 
n' and m being integers]; and so of other values of 7. 
These ‘independent variables” are therefore mutually depen- 
dent during their variation! Thus I have shown by an ex- 
ample,' which I thought too simple to need particularly in- 
stancing in my last paper, the fallacy of ohe of those principles 
which precede the application of P. Q.’s very elegant functional 
theorem, and have ‘therefore completely overturned the in- 
genious structure raised upon that principle. 
It will be recollected that I made no objection to the reason- 
ing in Mr. Herapath’s subsequent equations, so long as r and v 
were really independent variables ; but wished to show that as 
r and v were not independent variables in the case before us, 
the conclusions derived on the assumption of that non-existing 
independence were inadmissible as a demonstration of the 
binomial theorem. By tracing the process and finding that 
the independence of r and v was essential to the truth of those 
subsequent equations, I conceive that a complete neutraliza- 
tion was given to the evidence so obtained. 
The passage alluded to by P. Q. in his last paragraph cer- 
tainly was intended as an objection to Mr. Herapath’s mode 
of establishing some theorems in periodical functions, where 
the indices of the characteristic were fractional, that mode 
being founded on the assumed independence of two fractions 
whose sum is an integer. I have not the Number at hand, 
nor had I then; but I think I can depend upon my memory 
respecting the method. If I erred, let this circumstance apo- 
logize for me: but if I have not mistaken the method, and the 
preceding reasoning be admitted, the fallacy of such a method 
is apparent. 
I can assure Mr. Herapath (in conclusion of a reply which 
has expanded much further than I intended when I sat down 
to write), that were I convinced of the accuracy of his method 
I should “ not be backward to acknowledge it.” I trust I shall 
ever feel too sincere a regard for truth to contend upon any 
; question 
