714 
MR. R. C. ROWE OR ABEL’S THEOREM. 
In the second section (arts. 12-20) it is shown to follow from the results of the first 
that the sum of any number of integrals of the form considered may be expressed in 
terms of a definite number of such integrals, and the question what is the least 
value of this definite number is discussed : the result is stated at the end of art. 20. 
These articles correspond to pp. 211-228 in the original; they are rendered more 
direct by the nomenclature of ‘major terms’ and ‘sets/ the introduction of the 
letter r, and various minor changes of notation. 
Art. 21 contains an example of the method of this section. 
The third section contains two distinct parts : first, a generalization (art. 22) of the 
theorem of Section I., showing that a similar expression to that obtained there may be 
found for the sum of any number of such integrals each multiplied by any rational 
number positive or negative, integral or fractional; secondly, an investigation (art. 23) 
of the conditions necessary that the algebraic expression obtained for the sum of the 
integrals considered in Section I.— i.e., the right-hand member in the main theorem— 
may reduce to a constant. This article corresponds to pp. 196-208 in Abel, but the 
demonstration is greatly shortened and simplified by its being placed after (instead of, 
with Abel, before) Section II. 
Abel concludes by applying his methods to the case of integrals of the form 
I have succeeded in shortening the necessary work, but my process and result are so 
similar to those of the original as hardly to be worth reproducing here. 
An appendix contains an algebraical lemma and a list—it is hoped complete—of 
the errata in the original memoir. It appeared to the writer worth while to attempt to 
save subsequent readers the considerable inconvenience these errata had caused 
himself. 
There follows an addition from Professor Cayley, wdjerein it is shown that the 
expression found in art. 20 for the least value of the number of conditions connecting 
the variables of the integrals we sum is equal to the deficiency ( Geschlecht ) of the curve 
represented by the equation y(aq y) = 0. That this least value is equal to the deficiency 
is a leading result in Rjemann’s theory of the Abelian integrals ; the assumptions 
made in the text as to the form of the roots of the equation y(.r, y) = 0 considered 
as an equation for the determination of y are equivalent to the assumption that the 
curve x( x ) y) — 0 has certain singularities; and it is in the addition shown that the 
resulting value of the deficiency, as calculated by the formulae in Professor Cayley’s 
paper ‘On the Higher Singularities of a Plane Curve/ Quart. Math. Journ., vol. vii. 
(1866), pp. 212-222, has in fact the foregoing value. 
