We have seen in the ANALYST of Nov. and Dec. 1874, that the central equation of the ellipse may be written in this form (1) z = sin ( + ia). (e" - e-a). and the ordinate of The element of the The major axis of this ellipse is cos ia, or (ea + e-a) and the minor axis is (numerically, i. e., regardless of direction) sin iai, or The abscissa of any point of the ellipse is cos ia.sin the same point is (expressing its direction) sin ia.cos . area of the ellipse is therefore given by the expression A = sin ai.cos .d(cos ai.sin ) = cos ai.sin ai.cos2 d. Hence, integrating .cos + 5) (2) A.cos zi.sin ai(in which is the area of the ellipse from 0 to = ". This value of course appears here in an imaginary form, since according to our notation every ordinate is an imaginary quantity. Numerically we have (2a) A = [cos ai.sin ai (sin .cos + )] ÷ 2i which is a "real" magnitude. I will now show that the value of A in equation (2) is the same as that obtained by the ordinary method, namely, the central equation of the ellipse being a2y2+ b2x2=a2b2, we find x√ (a2 — x2) + a2.arc sin xv(a2 Introducing the eccentric angle, so that x= a.sin ; y=a.cos &; the last value of A becomes b Α A = 2a√ 1 a.sin 4. a.cos 4+ a2.4] = 'ab [sin 4.cos 4+4 which is identical with (2) for a = cos ai; b= sin ai (numerically). We now see that is the same as the "eccentric" angle . Let it be remembered that sin ai and sin ai i are identical so far as magnitude is concerned. The ordinary method does not express direction; but only magnitude. In the same number of the ANALYST I have shown that the central equation of the hyperbola may be written thus (3) z = sin (u+i). The axes are here respectively sin a and cos a; the abscissa of any point is sin a.cos i and the ordinate of the same point is, numerically, cos aX sin ii. The element of the area here becomes as the area of the hyperbola. This equation is precisely the counterpart of eq. (2). By the ordinary method we find from the central equation (4a) b2x2-y2a2=a2b2, for A the expression A = } [ xy — ab.log(2+ To compare this result with our own, I will again introduce the "eccentric angle", so that x= a.sec; y=b.tan &, then the last value of A becomes A=[a.sec.b.tan -ab.log(sec+tan )]. But since the abscissa x and the ordinate y according to our notation are xsin a.cos i; y = cos u.sin i÷i (absolutely) we find as necessary conditions that the two values of A become identical. For, now the last value of A becomes Aab[cos i.sin ii- log(cos i + sin ii)]. But cos i + sin i ÷ i = é; log (cos i + sin i÷i) = ; which shows that numerically (4) and (4.) are identical. For, the numerical value of A in (4) is A cos a.sin a.(sin ig.cos ii). = Equations (5) and (6) establish a simple relation between the circular functions of an imaginary arc in and those of a corresponding real arc ; and they give rise to a handsome construction, which leads us to understand the nature of the imaginary circular functions. Let PAP' be an hyperbola, whose central equation is x2b2-y2u2=u262; then, introducing the eccentric angle ACM . and drawing the tangent LM, we have x= a.sec&; y=b.tan. = & But from our equation of the hyperbola in connection with these last two equations we found the equations (5) and (6), which shows that, putting a=1, CL= CN; sin ini AN; cos in = = which last expression is of course again taken in the absolute sense, i. e., regardless of direction. And since cos pi dy These four equations are sufficient for a complete discussion of all the circular functions of an imaginary argument. Let it be observed that the argument in does not appear in the figure; but only its functions cos in, sin in and tan in which depend upon the angle . These four equations evidently hold for all values of and of x. Hence we have the following Theorem :-To every imaginary angle in, there corresponds a cognate real angle; and all circular functions of in can be expressed by means of circular functions of and vice versa. It is well known from the theory of complex functions that all relations obtaining for the circular functions of a real argument hold equally for the like functions of an imaginary and of a complex argument. Also that all the rules of differentiation and integration hold for imaginary functions. Hence we are warranted in deducing Nos. (7) and (8). If in the equation of the parabola we put a=b=1, then x2 — y2 = 1, and this is the central equation of the equilateral hyperbola. Again employing our imaginary function we have z=cos in sin in as the equation of the same curve. Now cos in being the abscissa and sin in the ordinate of this curve, we see that this curve gives immediately the values of cos in and sin in. A slight modification changes these two functions into what is well known as the hyperbolic cosine and sine of 7; namely sin ini sin hyp.ŋ. cos in = cos hyp.7; = But Gudermann, Lambert and their followers did not express the condition that cos.hyp.x and sin.hyp.x are perpendicular to each other (in our figure CN and AN) and thus they arrived at the fundamental equation (cos.hyp.x)2(sin.hyp.x)2 = 1 whilst our own notation gives us (cos is)2 + (sin i5)2 = 1 which at once reduces the so-called hyperbolic functions to circular functions. In fact this last formula together with the well known equation sin(ix iy) sin ix.cos iy± cos ix.sin iy = is sufficient to show that all the equations which hold for the circular functions of real arcs, are equally applicable to the circular functions of imaginary arcs. Now we have the following Theorem:-The so-called hyperbolic functions of a variable ŝ are numerically identical with the corresponding circular functions of i5; and all goniometric formulæ apply to the hyperbolic functions. The reason why the hyperbolic functions have not come into general use is, undoubtedly, the fact that formulæ expressing their relations to each other did not agree with the circular formulæ. I will now present two very convenient formula for the numerical computation of sin is and cos is. Employing the letter x instead of § we have, by expanding the exponential equivalent of sin ix a series which converges very rapidly and of which two terms afford all desirable accuracy at least as far as x = 1. In a similar manner we find By employing these two formulæ I have calculated the following little table of imaginary circular functions, which will be acceptable, since the tables of hyperbolic functions are not generally accessible. TABLE OF CIRCULAR FUNCTIONS OF AN IMAGINARY ARGUMENT. 1 1 1 1 Deg. Arc x cos ixsin irtan ix Deg. Arcx cos ixsin ix tan ix г 1 0.0174 1.0001 0.0174 0.0174 50 0.8726 1.4051 0.9877 0.70292 5 0.0872 1.0038 0.0873 0.0870 55 0.9599 1.4972 1.1178 0.74661 10 0.1745 1.0152 0.1754 0.1727 60 1.0472 1.6002 1.2494 0.78074 15 0.2618 1.0344 0.2648 0.2558 65 1.1344 1.7155 1.3939 0.81255 20 0.3490 1.0615 0.3555 0.3349 70 1.2217 1.8472 1.5491 0.83865 25 0.4363 1.0967 0.4503 0.4106 75 1.3090 1.9896 1.7179 0.86344 30 0.5236 1.1402 0.5478 0.4805 80 1.3963 2.1524 1.9005 0.88298 35 0.6108 1.1924 0.6495 0.5447 85 1.4835 2.3212 2.0928 0.90162 40 0.6981 1.2566 0.7562 0.6016 90 1.5708 2.5085 2.3002 0.91740 45 0.7854 1.3245 0.8683 0.6555 95 1.6581 2.7189 2.5283 0.92987 100 1.7453 2.9500 2.7757 0.94092 Before proceeding to the further application of our imaginary circular function, I will present two elementary integrals already known from the theory of hyperbolic functions. Putting arc sin ix = iy, we find These two formulæ will serve as elementary integrals precisely as their S1. +202 da arc tan x. For the integration of circular functions it is often quite advantageous to introduce the cognate imaginary angle. For example let the integral Introducing the cognate angle if, we have from (8) and (5) |