60 REPORT—1846, 
the latter is not, except under certain circumstances, soluble, though the 
solution of the equation on which it depends can be reduced to the solution 
of certain other equations of lower degrees. 
But for an infinity of particular values of the modulus, the case in ques- 
tion is soluble by a method closely analogous to that used by M. Gauss for 
the solution of binomial equations. Thus for all such values the problem of 
the division of elliptic integrals is completely solved. 
The most remarkable of these cases corresponds to the geometrical pro- 
blem of the division of the perimeter of the lemniscate. Abel discovered 
that this division can always be effected by means of radicals, and further, 
that it can be constructed by the rule and compass in the same cases (that 
is for the same values of the divisor) as the division of the circumference of 
a circle. Of this discovery we find Abel writing to M. Holmboe, “ Ah qu'il 
est magnifique! tu verras*.” 
In order to form an idea of the nature of the difficulty which disappears 
in the case of which we are speaking, let us suppose that we have to solve 
the algebraical equation which is represented by the transcendental one 
¢ (3 6) = 0, in the same manner as the equation 4.23 — 32 = 0 is represented 
by sin (36) =0. 
The roots of 4.23 — 3 7 = 0, are, setting aside zero, 
_ on . ar 
sin ——, sin —. 
3 3 
Those of the former algebraical equation, which, as we know, is of the ninth 
degree, are, beside zero, 
2w 40 
niga dg 
2ai 4 at 
kt Nanas 
—— —— 
Q(w+2ai) ~4(w+2ai) 
‘Sa ee 
where i= /— 1. 
To satisfy ourselves that these are the roots required, we observe that 
@ (mw + nai) =0 for all integral values of mand. Hence the general 
eanaae but it will be found that 
if we give any values not included in the above table to m and , the resulting 
expression can be reduced to one or other of the forms we have specified in 
virtue of the formula 9 (9) =o{(—1)™t"0+m wt+nai}. E.g. The non- 
5w+2at 4 (w+ at) 
3 
3 
form of the roots of our equaticn is ¢ 
, Since the 
tabulated root is equal to our sixth root ¢ 
sum of their arguments is 3w + 2a, and the sum of 3 and 2 is an odd 
number. 
* It is right to mention that M. Libri has disputed Abel's title to the theory of the di- 
vision of the lemniscate. I shall, however, not enter on the merits of the controversy which 
arose on this point between him and M. Liouville. The reader will find it in the seventeenth 
volume of the ‘Comptes Rendus.’ It appears that M. Gauss had himself recognised the 
applicability of his method to the equation arising out of the problem of the division of the 
perimeter of the lemniscate (vide Recherches Arithmetiques, § vii. p. 429. I quote from 
the translation published at Paris in 1809), . 
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