60 INFINITESIMAL CALCULUS whence, identifying, we have I = - 2a 3 , /= 2a, </=-2a 3 , and, , the theorem gives assumin another abscissa CE = s=- -- to " ~ - V/iar - ha 2 arc AB + avc AF = -"^ + K . V2 Taking another pair of abscissa; t, u, similarly related, we have tu^li T , arc Ac + arc Af /= + & > aV2a and by subtraction the arbitrary K is eliminated. 191. In order to be able to state the results more concisely, it is desirable here to explain in anticipation the notation introduced by Legendre, which has since generally prevailed. If the position of a point on an ellipse be expressed by the co ordinates x = asin<j>, y = bcos<p, it can easily be found that, denot ing the eccentricity of the ellipse by K, the arc reckoned from the extremity of the axis minor A to the point B determined by is expressed by arc AB /* /- - -^ =/ VI - K* sin 2 0d0. a Jo This Legendre writes arc AB = E(0). If it were desired to indicate also the quantity K, which is called the modulus of this elliptic integral, he writes it E(K, 0), and calls this an elliptic integral of the second kind, for reasons which will soon appear. The quantity is called the amplitude of the ellip tic integral, and its geometrical meaning is the eccentric angle measured from the axis minor of the point for which E(K, 0) measures the arc. For brevity also he adopted the notation Vf^lAin 2 ^ = A(K, 0), or = A(0) when it is unnecessary to mention K. 192. Legendre, proceeding to rectify the hyperbola -^ - -|^ = 1 , first assumes a; = asec0, and this gives the element of the arc but to have a radical similar to that of the dO = - /S 2 2 sin 2 arc of the ellipse he had recourse to another notation. ing by the equation Determin 11 = tan 0. where c 2 J c a? + j8 2 , we get x - aV 1 - /c 2 sin 2 COS in which OL = CK. The arc of the hyperbola is in this way found to be &- t~ d<j) c J cos 2 0A0 Again rf(tan0A0) = , wnere 2 + K"- = 1 or CK 2 tan 0, is AB = cA0 tan <f> COS 2 0A0 A0 Hence the hyperbolic arc AB of which the extreme ordinate is BD, -<j> ,< j. / A0cfy> + CK 2 / . o Jo A The geometrical meaning of is easily determined by taking the circle on the transverse axis, and joining its intersection with the tangent at B to the centre ; is the angle the joining line makes with the perpendicular on the tangent. It is seen thus that the arc of a hyperbola depends, not only on the integral which gives the arc of an ellipse, but also on /" ^ , which is called an elliptic in- Jo A< tegral of the first kind, and denoted by F(K, 0), and K bein called amplitude and modulus as before. 193. When this is applied to the formulae of Fagnani they become for the ellipse, calling ty the value which has for the point F, sin ^ = ^ , arc AB - arc GF = a* 2 sin sin * = a* 2 sin * COS A0 A0 and it is easily found that the expression on the right is the length on the tangent at either B or F intercepted between the curve and the foot of the central perpendicular let fall upon it. In application to the hyperbola, similarly, A0 and arc AB + arc AF = sm cos + const. Now the length of the tangent between the point of contact anc the foot of the central perpendicular upon it is c tan 0A0 ; hence denoting by E and F the complete functions E(K, ^ir), F(/c, ^TT), the value of the constant can be determined ; arc AB + arc AF A0 - + /c 2 F-E; c sin cos and this value of the constant is the difference between the entin hyperbolic quadrant and the length of the corresponding asymptote 194. Landen, continuing these investigations in 1775, consider; the hyperbola whose semi-axes are a=m -n and b = 2^mn ; then vriting CP= {m -n -t 2 }* (fig. 17), he notices that m+n - 1 2 Then he says it is well known that, in ilie cllipsc whose semi-axes ire m, n, the arc z from ^*> - he conjugate axis to the oint whose abscissa is x /"/2^,2 0-^*2 1 -=/ , V } dx, J m- - x- ) where g is nd therefore in the ellipse vhose semi-axes are m + n md 2/mn, for an ab- m + n t, the arc=f( m + n J m=n - Fig. 17. dt . But further, in the ellipse m, n, the length t of the tangent at the point whose abscissa is x, to the foot of the central perpendicular on it, is t--=gx ,-n -t" ni-n -fij J m + n - t- which is a relation between the hyperbolic arc, the two elliptic arcs, and a portion of a right line. Landen remarks on his dis covery, "Thus beyond my expectation I find that the hyperbola may in general be rectified by means of two ellipses," a result apparently of secondary importance compared with the method it is attained by, which involves the principle of what after Legendre is known as Landen s transformation. 195. But, simultaneously with the geometrical interest thus developed in these integrals, they also attracted attention to their more algebraic relationships. James Bernoulli (1694) devoted par ticular attention to the "elastic curve," which is defined by the equation dx= ?_-j^- , with a view to construct it by the quad- V a 4 - y 4 rature or rectification of a conic. Maclaurin (1742) gave such a construction, depending on the rectification of the equilateral hyper bola ; and in like manner constructed by the aid of arcs of conies the integrals of expressions such as dx dx and others like them, which can be reduced to elliptic differentials. D Alembert (1746) extended these results. His paper in the History of the Berlin Academy treats a number of differentials, whose in tegrals are of the same forms as those by which the arc of an ellipse or hyperbola is expressed. When a differential expression can be reduced to the differential element of the arc of one of these curves, D Alembert calls this the integration of it by means of an ellipse or hyperbola. This paper is of a purely analytic character, and is penetrated by a tendency to classification of elliptic differentials similar to that which has effected in the works of Legendre such important services in the development of analysis. 196. In the works of Enler, of which from 350 to 400 quarto pages are concerned with this department of our subject, the geo metrical and analytical aspects alternate. His first investigations are in the St Petersburg Commentaries (1761), on the integration of the differential equation mdx _ ndy Vl -a; 1 /l - j/ 4 and it is remarked that the differential equation of a more general form mdx __ ndy _ can be completely integrated by an algebraic equation, provided the numbers m and n are rational. He extends the same method of integrating to the apparently more general equation mdx ndy /A + 2B;r + G 2 + 2Dx 3 + EZ 4 ~ V1T2B1/ + C?/ + 2"Di/ 3 + El/ 1 In the next paper in the same volume Euler determines on the quadrant of an ellipse two arcs whose sum can be expressed geo metrically, reproducing many of Fagnani s formulae. 197. Euler s most important investigations are collected in his Institutiones Calc. Int. , vol. i. sec. 2, cap. vi. His method, being
