Having premised these lemmas, we proceed to the theorem where far is an integral function of a, and px = $ ̧x ⋅ P ̧x; p and a being also integral functions. (a, &c., and c, &c. being as heretofore new variables) we shall therefore also have is Hence dx φ = 2 F'x F'x fr [ { px (a ̧ +.....a „x”) (da ̧ +...+ xTM da„) - (co +...c, x") (de, +...+x"de„)}· Ò ̧x {(c + ... cx") (da, + ...x" da) (x − a) 、 px The coefficient of each of the differentials da, &c., de, &c. 1 (ra) F' multiplied by an integral function of a. On taking the sum therefore for all the roots of Fu = 0, we see from Leinma II., that dc„)} (da + ... u" da) − (a + ... au") (de + ...u" dc,)} -- 0 {(c + ... c,a") (da, + ... a" da,) Fu = q1u C2 - puÃ2, and the last written equation will become. Now a, fa, pa, p ̧a, u, fu, pu, pu are all constant, since they do not involve a or c, and a little attention shews that fПx(ux) dx = 1 sx(uz) dz, that is, that we can differentiate or integrate under the symbol П. and consequently integrating the last equation, and restoring the values of the different quantities which it involves, we shall have log (c ̧ + ...c„u")√p2u+ (a ̧+ ...a„uTM) √ p1 u (c + ... c,u") √ p2u − (a ̧ + ... a, uTM) √ p1u (c + ... c,a") √ p2a + (a ̧ + ... a„aTM) √p, a V which is Abel's theorem. น + C, In consequence of the ambiguity already more than once noticed, the signs of the transcendent functions a1, æ, &c. must be considered as hitherto undetermined, though not in reality indeterminate. If aa is a factor of fx, so that fr = (x − a) ƒ ̧x, where fa is an integral function, we shall have (c + ...cu") ✓ p2u + (a + ... auTM) √ p1 u for in this case fa = 0. + C', 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 I will disappear. And therefore in this case the general theorem will become (c + .... c„a”) √ p2a + (a ̧ + ama") φια ... ... (Co+... ca") v p2a - (a + ... ama") ν φια + C. The number of functions æ, æ, &c. under the symbol is of course that of the roots of Fu = 0. Of their arguments 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 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 L1 (1x) = log (1 ± a). 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 L1 (1±x), as it is a transcendent the properties of which are well known, let us take L, (1 ± x) = ƒda log (1 ± x). Using the lower sign, and changing 1 a into r, and into 1-æ, we have |