NOTICES AND ABSTRACTS 
OF 
MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 
MATHEMATICS AND PHYSICS. 
On Algebraic Equivalence. By the Rev. T. Jarrett, Professor of Arabic 
in the University of Cambridge. 
Many algebraic series, when applied to particular cases, have been found to in- 
volve numerical absurdities. Attempts have been made to explain these contradic- 
tions by means of metaphysical refinements, but in most cases without success. 
The object of the paper was to remind mathematicians that the series in question 
arise, in most cases, from a succession of operations, in each of which a portion of 
the result is neglected, and that’ the accumulated effect of these omissions shows 
itself in a numerical absurdity in particular cases. The writer took the binomial 
theorem as an instance, and showed in what way the demonstration would most 
naturally proceed, from the simple case of the index being positive and integral, to 
the cases of a negative integer, and of fractions both positive and negative. This was 
done by means of a notation* which expresses any series by its general term, and by 
means of which the operations of multiplication and involution can be as readily 
_ performed on a series as on a single term. ‘The result arrived at was, that no de- 
' pendence ought to be placed on, nor any use made of, the binomial theorem without 
the express limitation that either the index must be an integer, or that the second 
_ term must be less than the first. 
_ Many eminent mathematicians have explained the above-mentioned numerical 
absurdities, by making a distinction between algebraic and arithmetical equality, as 
founded on an assumed difference between symbolical and arithmetical algebra, and 
_ resulting in a separation between equivalence and equality. The writer called in 
_ question the justice of this distinction, and asked for information on a point so im- 
portant in estimating the truth and value of analytical investigations. 
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4 
z On Imaginary Zeros, c. By Professor Youne. 
The principal object of this paper was to remove certain contradictions involved in 
the prevalent theories of conjugate points, and to supply correct principles for their 
‘determination. These principles are thus announced :—1. Let the equation be solved 
for one of the variables; then, if any pair of values for x and y, which satisfy this 
ae equation, differ from real values only by the entrance of an imaginary zero— 
in whichever direction that zero he reached—those real values will be the coordinates 
of a conjugate point. 2. If it be inconvenient to solve the equation for one of the 
_ variables, we may take a differential coefficient of any order whatever: then, if a 
‘pair of values for x and y render this coefficient imaginary, and at the same time 
Satisfy the rational equation of the curve, those values will belong to a conjugate 
_ point. 
: The author further shows that it is equally correct to regard a conjugate point 
as an evanescent oval, or as a real point through which an imaginary curve passes : 
in the former view the differential coefficient, at the point, is indeterminate and real ; 
in the latter view it is determinate and imaginary. 
* A full account of this notation, with numerous applications to pure analysis, will be 
_ found in the writer’s ‘ Essay on Algebraic Development.’ 
— 1845. ° B 
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