Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/80

 70 INFINITESIMAL CALCULUS were subsequently called the 6 and &amp;lt;?j functions ; also appended to /9K ~* K - tliis, the development of. / by powers of q = c &, the expon ents of which are the squares of the natural numbers ; as also the very important development of V/c as quotient of two series pro ceeding by square powers of q and q$ results whose importance was at once accepted by Legendre. Regarding the result in this paper, that to a given modulus for a prime degree of transformation n there always correspond n + 1 other transformed moduli got by JL i A putting q n, q n , aq n ,. . . a n - l q&quot; for q, where a&quot; = l, Jacob! re marks, &quot;Thus M. Abel will see that imaginary transformations had not escaped me. &quot; 248. Jacobi s well-known construction fur the addition and multiplication of elliptic functions by the arcs determined on a circle by the vertices of an inscribed polygon, whose sides touch other circles coaxal with it (or, as he entitled it, application of elliptic transcendents to a known problem qf elementary geometry to find the relation between the distance of the centres and the radii of two circles, one inscribed in, and the other circumscribed to, an irregular polygon), is of about the same date, April 1, 1828. Immediately afterwards, Jacobi, still ignorant of the second part of Abel s &quot;Eecherches,&quot; communicates to Legendre (April 12, 1828), the forms of development detailed in the work we have just spoken of for the sin am, for the modulus of the integral, and for the period K, and notices that these formula will not be without interest for the celebrated geometers who are engaged with the motion of heat, numerators and denominators of the fractions by which the trigonometric functions of the amplitude have been expressed being often met with in that question. 249. Abel sought to generalize the problem of transformation, in the publication of which he was anticipated by Jacobi. &quot; We may regard this theory &quot; he says (May 27, 1828), &quot; from a much more general point of view, proposing as a problem of indeter minate analysis to find all possible transformations of an elliptic function which can be effected in a certain manner. I have attained the complete resolution of a great number of problems of this kind, among them the following: To find all possible cases in which we can satisfy the differential equation dy dx _ 8 &quot;&quot; : - ~- - by putting for y an algebraic function of x, rational or irrational. This problem may be reduced to the case that y is rational. For we can show that, if (1) holds for an irrational value of y, we can always deduce from it another of the same form, in which y is rational by suitably changing the coefficient a, the quantities c }, c t , c, c remaining the same. The fir^t method which presents itself for resolving this problem when y is rational is that of undetermined coefficients. But this is a very fatiguing process. The following, I believe, deserves the attention of geometers, leading as it does to a complete solution in the simplest manner.&quot; The theorem of the reducibility of the general problem of trans formation to the rational is, however, stated without proof in this paper, but the problem of rational transformation, based on con siderations of periods for the original and transformed elliptic func- tiOn, is strictly treated. It is shown to resolve into simpler analogous problems whenever the number characteristic of the transformation is a compound one, and the equation of transfor mation itself is stated to be algebraically soluble. Lastly, Abel enters more closely into the case of equality of the transformed moduli of the integrals (viz., c i = e, C] = e), which has subsequently constituted the theory of the complex multiplication of elliptic functions. The multiplier a of the transformation is found in the necessary form /* + J - p, where // and /.i signify two rational num bers, of which the latter must be essentially positive ; and Abel adds &quot;If we attribute to a such a value, we can find an infinity of different values of c and c which rentier the problem possible. All these values are expressible by radicals.&quot; Regarding the subject of this paper, Jacobi remarks to Legendre (June 14, 1829), &quot;Abel s principal merit in the theory of transformation consists in his demonstration that our formulee, embrace all possible algebraic substitutions, and this gives a high degree of perfection to this theory.&quot; 250. In the &quot;Suite des notices sur les fonctions elliptiques,&quot; dated July 21, 1828 (Crelle, vol. iii.), Jacobi introduces his functions and H as independent fundamental functions on which to base the theory of elliptic transcendents ; a conception to which also Abel vas simultaneously led, and which he gave utterance to in writing to Legendre, Nov. 25, 1828: &quot;The theory of elliptic functions has led me to consider two new functions which enjoy several remarkable properties.&quot; Abel desired, quite in analogy with Jacobi s principle, to treat of the properties of these new trans cendents apart from the inverse function of the elliptic integral, but the priority of publication of this discovery is Jacobi s, since the completion of the second part of the &quot;Precis d une thcorie des fonctious elliptiques,&quot; which was to contain all these investiga tions, was interrupted by Abel s unexpected death. The above-mentioned work by Jacobi next contains the theorems expressing elliptic integrals of second and third kinds by 6 functions. Regarding the formula of reduction of the integral of the third kind by aid of the functions, Jacobi remai ks a characteristic property to Legendre (September 9, 1828): &quot;Moreover it shows that elliptic functions of the third kind into which three variables enter reduce to other transcendents which contain only two,&quot; a discovery which Legendre was greatly interested in, though he found difficulties in the distinction of real and imaginary parameters, expecting that the introduction of an imaginary parameter involved three independent jquantities in the integral, and so there should be four kinds of elliptic functions instead of three. Jacobi, however, repeats the above assertion in his demonstration of the relation n(-M,) = Z + l og 0?^) (OeZfe.iv.): &quot;This {u 4~ a) latter formula shows that elliptic functions of the third kind which depend on three elements can be reduced to other transcendents which contain only two. &quot; Later, Jacobi wrote to Legendre (May 23, 1829): &quot;As regards elliptic integrals of the third kind with circular parameter, they do not admit of a reduction analogous to that of the logarithmic kind. In a general analytical sense not distinguishing between real and imaginary values, a formula embraces all cases ; but in applying to numerical calculation cases must be distin guished.&quot; And again, &quot;as to the numerical calculation of elliptic integrals of the third kind with circular parameter, I do not think you should too much regret the inconvenience that they cannot be reduced to tables of double entry.&quot; 251. The collected statement of his investigations, published by Jacobi as his Fundamcnta nova TJicorise Functiomun Ellipticarum in 1829, contains two main divisions, the first on the transformation of elliptic functions, the second on their evolution. We have already indicated many of the principles contained in this work, the most important of all being that of the double periodicity of these functions. As Jacobi says ( Works, i. p. 262) &quot;Elliptic functions differ essentially from ordinary transcendents. They have, so to say, an absolute manner of existence. Their principal character is to embrace all there is in analysis that is periodical. In fact, trigono metric functions having a real period, exponentials an imaginary period, elliptic functions embrace the two cases, since we have at the same time sin am (w + 4K) = sin am, sin am (u + 2i1^ ) = sin am ?/. Moreover it is easily demonstrated that an analytic function cannot have more than two periods, one real and the other imaginary, or both imaginary (complex) if the modulus 1; be so too. The quotient- of the periods of a proposed function determines the modulus of the elliptic functions by which it must be expressed by means of the 2K -- relations -- y, , 0), (the expansions for which in terms of q follow by 258). Perhaps it will be convenient to introduce this quotient ^- into analysis as modulus in piace of 7c.&quot; On these principles Jacobi subsequently founded a theory of hyperelliptic functions. 252. Jacobi s first evolution of elliptic functions is into infinite products, and is derived from the transformation from A to k, which is expressed by sinam(w, k) . /^ V* H sin am ( -f ~ M n and equivalent forms, by writing in the equations of transformation, for v, -*, and allowing n to take an n (u, Xj becomes - * ^ *=x, and he arrives at the equations mi ^i sin am&quot; 2k (l-2q _ A am _ 2q cos 2x + ^) (i _ 2q 3 cos 2x + g)~ from which are easily derived such series as 2/tK 2K.r 4 Vg&quot;sm a 4&amp;lt; r sinte 4A 5 sin 5.r - 1 n ~ = ._-_ + &c. __ .._- 1 - 2? cos 2x + q 2 1 - 2? 3 cos 2x