[712 
713] 
29 
gaiithin and 
nsider three 
ts 1, 2, 3 ; 
he equation 
t, the three 
the inter- 
my eight of 
713. 
ADDITION TO MR ROWE’S MEMOIR ON ABEL’S THEOREM. 
[From the Philosophical Transactions of the Royal Society of London, vol. 172, Part in. 
(1881), pp. 751—758. Received May 27,—Read June 10, 1880.] 
In Abel’s general theorem y is an irrational function of x determined by an 
equation ^ (y) = 0, or say % (x, y) = 0, of the order n as regards y : and it was shown 
by him that the sum of any number of the integrals considered may be reduced to 
a sum of 7 integrals ; where 7 is a determinate number depending only on the form 
of the equation y (x, y) = 0, and given in his equation (62), [Œuvres Completes, (1881), 
t. I. p. 168] : viz. if, solving the equation so as to obtain from it developments of y 
in descending series of powers of x, we have* 
Vl\ 
Hj/ij series each of the form y = af-' + ..., 
n 2 p 2 
y = of* + ..., 
WjbMfc 
y = + 
* The several powers of x have coefficients: the form really is y = A l x >L ' + ..., which is regarded as 
l 
representing the ¡x x different values of y obtained by giving to the radical each of its values, and 
the corresponding values to the radicals which enter into the coefficients of the series: and (so understanding 
it) the meaning is that there are 11 s such series each representing ¿4 values of y. It is assumed that the 
l 
series contains only the radical x 1 *', that is, the indices after the leading index 1 are 
Jilj - 1 9JÌJ - 2 
, ... ; a 
Mi Mi Mi 
series such as y = A 1 x* + B 1 x% +..., depending on the two radicals x x , x° represents 15 different values, and 
would be written y=A 1 x^% +..., or the values of m 1 and y x would be 20 and 15 respectively: in a case like 
this where — is not in its least terms, the number of values of the leading coefficient A, is equal, not to 
Mi 
H 1 . but to a submultiple of . But the case is excluded by Abel’s assumption that —, are fractions 
each of them in its least terms.