64 an additional complication. Legendre first establishes the relation -p + 1 , V a tan where a = (l + by means of which any function n having a parameter greater than c is reduced to depend on one having a parameter less than c, but with the same amplitude and modulus. The quantity a, however, may have different values, and thus the following cases are to be distinguished. When a is positive (either n positive, or if negative its value is between - 1 and - c 2 ) the function introduced is circular, as written above. When a is negative, n is negative, and either greater than - 1 or less than - c 2 , and the function is logarithmic. In this case writing a= - /3, the comparison written above is 2N/B VA- V tan 0. When a = the integrals are expressed by the first and second kinds. Omitting the case of ?i= - cosec 2 0, which can be reduced to that of 11= - c 2 sin 2 0, this case and those of n = cot 2 and n= - 1 + b* sin 2 remain, the first being the logarithmic parameter. The other two cases are easily shown to be connected by the relatipn ll l n(*) -i^n( -m) = +-^= tan-i V*Bj^in_0cps_0 m mn f mn A provided (1 + n) (1 m) = b~, and so constitute really but one case. Functions with imaginary parameters always reduce to others with real parameters of the above two distinct kinds. 211. Comparison of integrals of the third kind by means of the addition theorem leads to the formula
- -
V a l+n-n cos /j. cos cos j/ and thus the difference, which is zero in the first kind, and is algebraic in the second, is here expressed by the arc of a circle ; which becomes a logarithm if a be negative. Thus finally Legendre remarks that if where P is a rational function of x, there can always be found an. algebraic equation between x, y, z, &c., such that the quantity where i, /:, I, &c. , are integers, may be determinable by arcs of circles and by logarithms. 212. Legendre next proceeds to the discovery of Landen, having so far been employed mainly with those of Euler. He expresses astonishment that among the many analytic transformations em ployed by Maclaurin and D Alembert they had not fallen in with the transformation which brings to light the numerous properties of the chain of moduli, and that this discovery was reserved for Lan den, who, however, made but a poor use of it, not even seeing that it furnished a very simple method for approximate calculation of the arcs of conies. It is less astonishing that Euler missed this discovery considering that the beautiful integration which is due to him led him to compare together the different valuesof the same transcendent, just as arcs of the same curve are compared. But nowhere in his Mtmoircs do we find him varying the constants or parameters of the functions, and thus passing from one curve to another, as is done in comparisons which depend on the scale of moduli. From the fact that Euler has written nothing about the memoir of Landen, Legendre concludes he had never been acquainted with it. 213. The formula given in 194, by introducing the eccentric angles from the axes minor in the two ellipses, easily gives rise to the equation sin = (1 + 6) sin A COS -. < ^ where 6- , and so _ A m c = V# of that article. belongs to another ellipse, and for it the value of the corresponding modulus is evidently - . Legendre sees through the simple proposition of Landen, expressing the arc of a hyperbola by two arcs of ellipses, to the infinite series of moduli which can be produced by repeated application of this sub stitution. Commencing with functions of the first kind, he shows that F(c,0) and F(c ,0 ), when c = - - c , and is determined by J. ~T" C >sin (2<J> -0) = c sin 0, are related by the equation c, 0). Thus, as20 -0 is always contained between + 6 and - 0, 6 being the least arc having < for its sine, there is no ambiguity in doter- mining the values of and <t>. The relation for complete functions is F 1 (c ) = (l +c)F 1 (c). Now conceive an infinite series of increasing moduli it will soon attain the limit unity. Forming the complements b , b", &c., of these moduli, the series decreases continually, and each term is, according to the law, & = !-- b" = l ~ C b "- 1 -" " &c The scries of amplitudes is deduced in succession by the formula; sin (20 -0) = c sin 0, sin (20"-0 ) = c sin , sin (20 " - 0") = c" sin 0", &c. ; and so a series of integrals of the first kind is got, related as follows F(c", 0") = i-T_ F(c , ) = if: 1 7 C F(c, any two of the functions being always in a ratio independent of the values of the corresponding amplitudes. Similarly for the com plete functions FV) = (1 + c)FV), FV) = (1 + c )F V) = (1 + c)(l + c )F ] (t:) , &c. But the series c, c , c", increasing in one direction, can be pro longed indefinitely in the opposite or decreasing sense to the limit zero. Here the law of terms is 2 c^ 2 c^ 2 x//j^ofl c = q n c = J -^., c==- m , &c. ; 1 -4" C 1 -4- i 1 -i- r and similarly 1-5 1-6 with the series of relations of amplitudes derived from sin (20 - 0") = c sin 0, &c., which may be written tan (0-0) = & tan 0, &c., n i ,.0 and of integrals F(c, 0) = ^ - F(c, 0) , &c. ; 2 2 or, reversing, since 1 + c = - , F(c, 0) = (l + J)F(c, 0), F(c p , 00 ) = (l+i)F(c , 0) and, for the complete integrals, for the decreasing scale of moduli. 214. Now if this be applied to the second kind of integrals we find 6 2 F(c, 0) = 2E(c, 0)- 2(l + c)E(c , ) + 2csin 0, showing that an integral of the first kind can be expressed by the aid of two arcs of ellipses, E(c, 0) and E(c , ). Whence by the formula of 192 it follows that an arc of a hyperbola can alwajs lie expressed by two elliptic arcs, the theorem Landen enriched geometry by. Also, by eliminating the integral of the first kind by means of two consecutive equations in the series, the relation between the arcs of three consecutive ellipses in the series can be found ; so that by the indefinite rectification of two ellipses in the series . . . E(c", 0"), E(c , ), E(c, 0), E(c, 0), E(c, on ), . . . of which the extremes are the ellipse having an eccentricity 1, which is a portion of the axis major, and that having an eccentricity 0, which is a circle, the rectification of all the rest is obtained. The transformations of Lagrange, or of Gauss ( 203), may be seen to be essentially the same as this of Landen (or Legendre), for by taking sin <f> = i T / . ., / in F(c, 0) we get 2/fc" , where c = - , , or& = fl. sin 2 v// 1 + * Vl - c 2 sin 2 Vl-, Hence F(o, 0) = (1 +c)F(c n ,4<). Now with F(c n ,^) = iF(c,0), F(c, 0)= l .^F(c,<f>) , and by elimi nating the quantity L between the equations sin = ^ ^-L i^i . 1 + c" sin -i|/ tan = tan ^ Vl - c 02 sin 2 ^, we obtain the relation given above sin(20-0) = c sin 0. 215. This principle of transformation is next applied to the ap proximate calculation of the three kinds of integrals. Required, for instance, an approximate value of F(c, 0) : the decreasing moduli
