C welts: 
XV. Reply to Mr. Davies's Postscript on Mr. Herapatn’s 
' Demonstration. By P.Q. 
To the Editor of the Philosophical Magazine and Journal. 
Sir, 
I BEG to trouble you with one more view of the extremely 
simple and obvious matter of dispute between Mr. Davies 
and myself; and if this be not satisfactory to Mr. D. I must 
despair of giving any that is, and shall therefore give up the 
point. 
Let z,y be any two variable and absolutely independent 
integers; and let « be any non-integer also-independent of 
both wand y. Then 
(x + a) + (y — a) =z = some variable integer. 
But because w is a variable integer independent of y, and a is 
independent of both, x + a considered as a variable, whose 
changes are by integral saltations, is independent of y — a, 
whose changes are likewise by integral saltations. That is, the 
saltations of x + a do not necessarily affect the value or salta- 
tions of y — a; which is evidently the utmost that the princi- 
ple of Mr. Herapath’s demonstration requires; for it only 
needs that the function of one non-integer, of z + a for ex- 
ample, should not be affected by the changes of the other, 
y—a. Mr. Davies’s error seems to consist in making the 
variation of these non-integers continuous and to depend on 
that of the common part a. 
I wish to make no observations that may provoke a reply; 
but I beg to observe, that Mr. Davies is altogether mistaken 
in saying, “ that the inquiry does not call for the considera- 
tion of indeterminate integers.” The spirit of Mr. Herapath’s 
demonstration (Phil. Mag. for May 1825, p. 324,) is expressly 
founded on the assumed indeterminate nature of the integer 7, 
as any one may see by referring to the above page. 
If Mr. Davies will also turn tothe page he refers to, “ Phil. 
Mag., November 1824, p. 333,” and also to Annals of Philos. 
for December 1824, p. 420, he will find that “ Mr. Hera- 
path’s mode of establishing some theorems in periodical func- 
tions,” is not ‘* founded on the assumed independence of two 
fractions whose sum is an integer;” nor has it the most di- 
stant allusion to such a principle. Of course, Mr. Davies’s 
objections, which he admits are grounded on such a presump- 
tion, rest on mistaken views. 
I have already observed, that I wish not to say anything 
which may educe a reply; therefore, and therefore only, I pass 
over 
