[ 751 ] 
Addition to Mr. Rowe’s Memoir. 
By Professor Cayley, F.R.S. 
Received May 27,— Read June 10, 1880. 
In Abel’s general theorem y is an irrational function of x determined by an equation 
x{u )= 0 (or say y(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 y 
integrals; where y is a determinate number depending only on the form of the equa¬ 
tion x(x, y) = 0, and given in his equation (62) p. 206: viz., if (solving the equation so 
as to obtain from it developments of y in descending series of powers of x) we have* 
n 1 /x 1 series each of the form 
771 <j 
n 2 y, „ „ y—xF* 4- . . . , 
niyu 
y — x^ + 
(so that 7L = n ] iJL l d~n. : fx i . . . -\-nyit), then y is a determinate function of n 1; m ]5 y 1 ; 
%, m 2) y-2 ’ • • • n fo m h yic- 
Mr. Rowe has expressed Abel’s y in the following form, viz., assuming 
m n m o fflj 
— >—■’ • • • >— , 
i AO AU- 
* The several powers of x have coefficients: the form really is y=A 1 a> 1 + . . . , which is regarded as 
representing the /q different values of y obtained by giving to the radical each of its /q values, and 
the corresponding values to the radicals which enter into the coefficients of the series: and (so under¬ 
standing it) the meaning is that there are ?q such series each representing /q values of y. It is assumed 
1 /yyi 772 X 
that the series contains only the radical aid, that is, the indices after the leading index — 1 are —--, 
f'l /h 
772-1—2. 
———, . . . ; a series such as ^=A 1 al+B 1 a°+ . . . , depending on the two radicals ad, xi represents 15 
different values, and would be written y=A 1 %ii+ ..., or the values of and /q 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 x is equal, not to /q, but to a submultiple of /q. But the case is excluded by Abel’s assump¬ 
tion that —, — . . . , are fractious each of them in its least terms. 
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