2 REPORT—1846. 
4 
dered convergent instéad of periodic, and the integral is rendered determinate instead 
of indeterminate. 
To avoid the recurrence of these errors, it is proposed to divide infinite series and 
definite integrals into two classes, those which are dependent upon some condition of 
continuity, and those which are altogether independent, or neutral. Hutton restricted 
the term neutral series to the form 1 — 1 + 1 —14, &c., because of its being neither 
convergent nor divergent. It is here proposed to extend the signification of this 
term, so as to have no especial reference to convergency or divergency: a strictly 
neutral series may be either convergent, divergent, or periodic. 
Some controversy has arisen of late respecting Poisson’s theory of definite integrals, 
and certain forms have been condemned as erroneous which are really correct. Thus 
the integrals ; 
% sin ax 2 COs ax 
Eis z dwand J 9 Tha dx 
have been recently affirmed to be indeterminate, which they arenot. The second of 
these however has been investigated by Legendre, Gregory and others, by methods 
altogether objectionable, as is fully shown in the present paper. A correct process 
for obtaining the proper determinate result, has been given by Sir W. R. Hamilton 
in his paper on Fluctuating Functions, in the nineteenth volume of the Transactions 
of the Royal Irish Academy. The ordinary investigations of the first of the pre- 
ceding forms are correct, although objected to in a paper in the recently published 
per of the Cambridge Transactions, yet the conclusions obtained by Euler, Fourier, 
oisson, and indeed by analysts generally, in reference to this integral, are affected 
F Cy : 
with error: the values of the integral are always stated to be A ops 2 according 
as the constant a is positive, zero, or negative. 
It is easily shown however, by a reference to the law of continuity, that the 
middle one of these values, viz. 0, has no existence; for if « become zero by vanish- 
ing positively, the value of the integral is still 7 ; and if it become zero by vanishin 
8 Pp y: 8 5 y g 
negatively, the value is — a i 
Among the collateral topics discussed in the present paper, notice is taken of the 
method proposed by Deflers, and so often quoted by Poisson, for verifying the well- 
known integral theorem of Fourier; this method has been properly objected to by Mr. 
De Morgan, as involving an inadmissible principle: by.a little modification, sug- 
gested by the theory unfolded in this paper, the defect is removed, and Deflers’ short 
and ingenious proof of Fourier’s remarkable theorem rendered conclusive. 
The paper terminates with some observations on what is called discontinuity, a term 
which it is thought is often injudiciously and unnecessarily employed in analysis. 
It is suggested that expressions called discontinuous may generally be contemplated 
with advantage, as consisting of distinct continuities embraced in a single form. An 
instance of this is shown in the consideration of definite integrals of the form 
A tm a which are treated by Poisson, ‘in the eighteenth cahier of the Journal of 
—™m « 
the Polytechnic School, but whose conclusions are, by this mode of viewing the — 
integral, shown to be erroneous. The entire paper will probably be published in the 
Cambridge Transactions. 
Letter, on the Deviation of Falling Bodies from the Perpendicular, to 
Sir Jonn Herscuet, Bart., from Prof. OERsTED. 
The first experiments of merit upon this subject were made last century, I think in _ 
1793, by Professor Guglielmini. He found in a great church an opportunity to 
make bodies fall from a height of 231 feet. As the earth rotates from west to east, 
each point in or upon her describes an are proportional to its distance from the 
axis, and therefore the falling body has from the beginning of the fall a greater ten= 
dency towards east than the point of the surface which is perpendicularly below it; 
thus it must strike a point lying somewhat easterly from the perpendicular. Still, the i 
P b'| 
i 
