INFINITESIMAL CALCULUS 65 c, c, c 000 , . . . must be calculated, and the increasing amplitudes 0, 0, 000 , . . . thus F;c, -rt-^V, 0) = ^p ^V , *>) = &c. ; but, when the modulus has become very small, A = 1, and / -? = 0; if then * be the limit of the angles J<, J0, i0 000 , &c., we have F(c, </) When = i7r the limit * will be equally |ir ; so that the complete function is The continued product which multiplies TT, or *, may also be written in a form suited to logarithmic calculation, as The moduli are best got by taking auxiliary angles: let sin /i=c, .-. c = tan 2 ^, similarly if c = sin/* , c - tan 2 J/*, &c. ; and when a very small c has been arrived at, wo can get the next by Also the angles (f>, </>, are best found by tan (<f>- <) = & tan <p, taking for - (f> not always the least angle given by the tables but that which is nearest (p. 216. Combining the equation n = /^ $-r. with that de rived from it by differentiation with regard to n, and using a to denote (1 + n) ( 1 + ), as in 210, it is easily found that V n J dn -ivfdn , fdn Applying this to the case n = cot 2 0, and writing for brevity cot 2 = n , the following relation is found : sin cos sin $ cos < = ^ir + tan 0A(5, 0)F(c, <) + tan <A( + F(c, <)F(5, 0)-F(c, <^>)E(5, 0)-E(c, <p)F(b, 0). Making <f> = |TT, this gives for the complete function of the third kind, with positive parameter, the following expression : . A(M) sin cos 6 0)F(c) A similar relation is established for the other cases of the para meter, and in each the complete integral is likewise expressed by integrals of lower kinds. There now follow the general reduction of integrals with imaginary parameter, and the reduction to elliptic integrals of integrals not included in the general type, as for instance r dz r d<f> r d<f>
217. In his preface, however, Legendre had directed attention to the discovery of a new scale of moduli, different from that hitherto known, as the most novel of the results distinguishing this work from his Excrdces. This transformation starts from the assumption and by the conditions that and <f> reach ^ir together, and, moreover, that cos a does not contain any other irrational factor in sin </> but cos <, we get Now tan 1 + k sm~</> But in order that u should increase "gradually from to ir, as <j> does, h must be less than 1, and m less than 3. Again, if c and a are two moduli, so related that 1 - a 2 siu 2 a) = (1 - c 2 sin 2 d>) I . , . r 1 + K sm it will be found possible in general to satisfy the above equation, and thus we get -16m* whence m must be between 3 and 1 in order that a and c may both be real proper fractions. Hence and this, combined with the above differential relation, gives da md<p A(a, on) A(c, <j>) or, integrating, F(a, eo) = m(c, 0), a relation between two functions of the first kind, whose moduli depend in general on the quantity m, which may be taken at will between the limits 1 and 3. The modulus, a is always greater than c, for we have c m 2 -TO TO + 3 and m + 3 - < which is always positive. "We have seen that a and c are deter mined by means of the regulator m when it is known ; it can be found from either of them by solving a biquadratic. Again, the complements of the moduli are found by , = (m + l)(3-m) 3 whence follows the simple relation 16m Application of this transformation to integrals of the second and third kinds gives rise to the remark that the trisection of an indefinite function of the first kind may be reduced to depend on the solution of two cubic equations. 218. Now, starting with a given modulus c, an infinite series of moduli increasing towards the limit 1, and an infinite series decreas ing towards the limit 0, may be formed, and we may denote the latter by a notation analogous to the former. Let them be in the increasing order c, c lt c 11( c nl , &c., and in the decreasing order c> c , C 00 , &c. ; and similarly for the complements, the regulators, and the amplitudes. Thus, by the first scale, any integral of the first kind, having a given modulus and amplitude, can be trans formed into another with any modulus in the series . . . c, c, c, c , c". . . and from this by the second to any in the other series formed from the same c by a different law, depending on extrac tions of square and cube roots. Legendre arranges the moduli in a sort of infinite chess-board, having c in the centre, and the moduli derived according to each scale in rectangular directions, and notices how remarkable is this infinite multitude of transformations which the same function F(c, <j>) may be submitted to, without changing its nature while preserving the same ratio between the new function and the old for all values of the amplitude ; in vain, he adds, might a second example be sought of a function which should be reproduced under so many different forms, and to which, more justly than to the logarithmic spiral, might be applied James Bernoulli s device, "Eadem mutata resurgit." 219. The first volume of the Traiti also contains the reduction to elliptics of a great number of integrals, the development of elliptic integrals in series proceeding by sines and cosines of multiples of the amplitude, and calculations of some definite integrals, single and double, which can be expressed by elliptic integrals. The applications are, in geometry to the surface of an oblique cone, to that of an ellipsoid, and to a geodesic on a spheroid ; and in mechanics, to the rotation of a solid, to the motion of a body under the attraction of two fixed centres, to the attraction of homogeneous ellipsoids, and to the orbit described under a given central force. The second volume contains details of the calculation of the integrals, and such tables of them as have to be constructed in order that the use of these functions may be introduced into analysis just as circular and logarithmic functions are employed. Here, Legendre excludes the thought of reducing to tables functions of the third kind, since they contain besides the principal variable two arbitrary quantities ; and so the tables should be of triple entry, a thing altogether unmanageable. Besides these, this volume contains a treatise on Eulerian integrals, and an appendix on spherical functions and on quadratures. The third volume of the Traite contains three supplements to the theory of elliptic functions, dated 1828, 1829, 1832, embodying Legendre s acceptance of the discoveries made by Jacobi and Abel since the publication of the Traiti. 220. It was owing to the strangeness of his subject that Legendre for more than twenty years found no fellow-worker in it. "After having employed myself for a great number of years," he says in the preface to the first supplement of the Traiti, "with the theory of elliptic functions, of which the immortal Euler had laid the foundations, I thought I should collect the results of this long work in a treatise, and this I published in the month of January 1827. Up to that geometers had taken almost no part in this kind of researches ; but hardly had my work seen the light, its name could hardly have become known to scientific foreigners, XIII. 9
