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XY1I. Memoir on Abel’s Theorem. 
By K. G. E,owe, M.A., Fellow of Trinity College, Cambridge. 
Communicated by A. Cayley, LL.D., F.R.S., Sadlerian Professor of Mathematics in 
the University of Cambridge. 
Received May 27,—Read June 10, 1880. 
The object of this paper is to present in a shortened and simplified form the processes 
and the results of Abel’s famous memoir ‘ Sur une propriety generale d’une classe tres- 
etendue de fonctions transcendantes,’ composed and offered to the French Institute in 
1826, but first published in the ‘Memoires des Savans Strangers’ for 184l(pp. 176-264). 
The generality and the power of this memoir are well known, but its form is not 
attractive. Boole indeed in a paper on a kindred subject (Phil. Trans, for 1857, 
pp. 745-803) says: ‘'As presented in the writings of Abel . . . the doctrine of the 
comparison of transcendants is repulsive, from the complexity of the formulas in which 
its general conclusions are embodied.” Boole’s theorems however escape this charge 
only with loss of the generality which makes Abel’s valuable. 
But this complexity is rather apparent than fundamental. It is here attempted, by 
re-arrangement of parts, by separation of essential from non-essential steps, by changes 
of notation, in particular by the introduction of a symbol and a theorem discussed by 
Boole in the paper already referred to and by the addition of examples of the pro¬ 
cesses and results, to reduce this part of an important subject to a shape more simple, 
while no less general, than the original. 
The first of the three sections into which the following paper is divided contains 
(arts. 1-10) the investigation of the principal theorem of Abel’s memoir : these articles 
correspond to pp. 176-196 of the original, but are much simplified by the aid of 
Boole’s proposition : the theorem is written at the end of art. 8 in the form 
and answers to Abel’s equation (37), p. 193. 
In art. 11 three examples are given of Abel’s theorem. Those have been chosen of 
which the results were well known (e.g., the circular and elliptic functions) with a 
view to the comparison of this and less general methods.* 
* For other methods of solution compare Leslie Ellis, B.A. Reports for 1846, p. 38; Legendre, ‘ Fonc. 
Ell.,’ t. iii., p. 192; Boole, loc. cit., arts. 18, 24. 
MDCCCLXXXI. 
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