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Alico com o • (--.*-1, 67-94-
Lastly and Desert Fu - - - - o as before.
Having premised these lemmas, we proceed to the theorem itself. Consider the integral
y = fora dv,
(x – a) v 0x where fx is an integral function of X, and px = 0,8.08; dix and 02x being also integral functions.
Now a, fa, pia, 0,a, u, fu, diu, pou are all constant, since they do not involve a or c, and a little attention shews that
SIIx(uz) dz = [1 /x(uz) dx,
that is, that we can differentiate or integrate under the symbol n.
and consequently integrating the last equation, and restoring the values of the different quantities which it involves, we shall have
In consequence of the ambiguity already more than once noticed, the signs of the transcendent functions fan, Voxx, &c. must be considered as hitherto undetermined, though not in reality indeterminate.
If x-a is a factor of fx, so that fx = (x – a) fix, where fix is an integral function, we shall have
Again, if the index of the highest power of u in fu be less than half the corresponding index in pu, the term affected by the symbol  will disappear. And therefore in this case the general theorem will become
The number of functions VXi, yox, &c. under the symbol is of course that of the roots of Fu = 0. Of their arguments xi, 1.2, &c. a certain number may be considered independent variables, namely, as many as there are disposable quantities a and c, or m +n + 2. When a and c have been suitably determined in terms of the independent arguments, the other arguments will be given as the roots of an equation whose degree is less than that of Fu = 0 by m + n + 2.
It will assist the student in forming a distinct conception of Abel's theorem, to consider it as a result of the same character as the simple examples with which we set out. It differs from them merely because the assumption made is much more general.
Full developements of the theory of elliptic functions will be found in Legendre, Théorie des fonctions elliptiques, in Jacobi, “ Nova Fundamenta, &c.", and in the works of Abel. The work of Professor Verhulst, published at Brussels in 1841, contains, in a condensed form, the principal discoveries of Legendre and Jacobi, and will probably be found useful. It contains also some original matter, which is not without interest. There are also many memoirs in Crelle's Journal, both on elliptic functions, and on those which are called hyperelliptic or Abelian. We may refer also to a paper by Ivory in the Phil. Trans. for 1831.
Spence, in his Mathematical Essays, has given the name of Logarithmic Transcendents to functions of which the general form is
which he denotes by the characteristic symbol
L, (1 + x). It is easily seen that when n = 1 the series is that of the logarithm of 1 + x, according as the upper or lower sign is taken, so that
L (1 + x) = log (1 + x). All these transcendents, including the logarithm, may be expressed by means of integrals which have a mutual dependence on each other. Thus
From these integrals various properties of the transcendents may be deduced by analytical transformations, some of which are here given.
(7) Omitting L (1 + x), as it is a transcendent the properties of which are well known, let us take
L, (1 + x) = 50;" log (1 + x). Using the lower sign, and changing 1 - x into ir, and ir into 1 - X, we have
L2(x) = 51.4. log (a).