MEMOIR OF LEGENDRE 153 



ent manner, a part of the theorems with which they are filled to profusion.* 

 But, in the publications of 1786, remarkable as ihey were, these rich materials 

 hardly yet formed a completed edifice, and M. Lcgendre was not long in per- 

 ceiving that this subject, and in general the theoiy of transcendents wdiose dift'er- 

 cntial enters into the form above indicated, required to be treated in a manner 

 more methodical and thorougii. This he undertook to do in a Memoire sur les 

 transcendentcs cHiptir[acs, read by him to the Academy of Sciences in April, 

 1792, and published towards the end of 1793, in which he proposed to compare 

 among themselves all the transcendents in question, to class them according to 

 their different kinds, to reduce each of them to the most simple form of which 

 it is susceptible, to estimate their value b}^ approximations the most prompt and 

 facile ; and, in line, to form from the collective theory a sort of algorithm ;ivhich 

 should serve to extend the domain of analysis.t 



Taking, in its most general algebraic form, the differential already indicated 

 as a point of departure for this kind of researches, he analyzes it with extraor- 

 dinary address, lays aside all the parts which are integrable, whether ])y arcs of 

 the circle or logarithms, and thus reduces it to its quintessence; that is to say, 

 to the parts whose integrals are transcendents of a superior order. Then, trans- 

 forming this remainder by means of circular functions, he reduces it to a form 

 of wonderful simplicity, containing but five quantities :| an arc of the circle 

 designated by the name of an)pHtndc, null at the point where the integral com- 

 mences, and developing itself in proportion as that is extended; a modidus 

 always real and smaller than the unit, which, in the case Avhen an ellipsis is in 

 question, represents its eccentricity; ajMramctcr of any magnitude, positive or 

 negative, capable of being reduced to zero, but to which it would be useless to 

 attribute imaginary values ; lastly, two coefficients Avliose values, independent 

 of all the rest, may be anything, provided the}' be not null simultaneously. 

 The amplitude is the variable in relation to which the integration is made ; it 

 is null only at the point of departure from the integral. Tiie modulus cannot 

 be null without the expression being completely altered in its nature, but the 

 three other quantities may be null independently of one another, or fulfil in 

 their relations of magnitude certain conditions according to which elliptic tran- 

 scendents are divided into three classes. 



The second class is the only one which represents arcs of the ellipsis. The 

 first class is a transcendent more simple than arcs of the ellipsis; it may itself 

 be expressed by means of such arcs, but an arc of the ellipsis cannot be 

 expressed by transcendents of this first class. The third class, on the contrary, 

 the only one in which the parameter is not null, is more composite than arcs of 

 the ellipsis. 



The gradation which exists in the complexity of these three classes of tran- 

 scendents is manifested especially by this circumstance, that transcendents of 

 the first species may be joined Avith one another, by addition and subtraction, 

 so as to form a sum constantly null. Transcendents of the second species may 

 unite in like manner, so as to form a sum whose value is expressed in terms 



* "I shall not conclude this article," (XVI of the memoir,) says M. Legeiidre, "without 

 f:fiving notice that the greater part of the propositions contained therein have been discovered 

 by M. Euler, and published in the 7th volume of the Noiircaux Mimuiris de f'ttersh urg 

 and in some other works, a fact of which I was ignorant when I was engaged in these 

 researches. But the difference of the methods may throw new light on this subject, and 

 moreover the comparison of the arcs of different ellipses, which is discussed in article XIII, 

 bas not, as far as 1 am aware, beeu before treated of by any one." — Mem. l\4cad, des Scien- 

 ces, 1786, p. 676. 



+ Legendre, Theorie des fonctions elliptiques. Introduction, p. 3. 



t For this he employs the following expression : 



J 'A -\- B sin "d) d 6 



-f II siu '<p i/ I — «■•' sia -^^ 

 -Memoires sur les transccndantes tlliptujues, p. 17. 

 L 2 



