Gudermannian Complex Angles

VOL. 14, 1928

ENGINEERING: A. E. KENNELLY

839

GUDERMANNIAN COMPLEX ANGLES By ARTHUR E. KBN'NI4LLY ENGINZXRING SCHOOL, HARvARD UNIVIRRsITY Commnunicated October 4, 1928

It is well known' that corresponding to any real, positive hyperbolic angle 0 (hyperbolic radians) there exists a real positive circular angle 18 (circular radians) defined by the relations: (1) sin 13 = tanhO (2) cos 13 = sech 0 = 1/cosh 0 (3) tan ,=sinh 0 (4) cot 13 = csch 0 (5) sec 13 = cosh 0 (6) csc 13 = coth 0. This circular angle 13 is commonly called the "gudermannian" of 0 and the relation is expressed by the two well-known real equations: 13 = gd 0 real circular radians (7) and 0= gd-'13 real hyperbolic radians (8) so that the hyperbolic angle 0 is the antigudermannian or lambertian of 13. Gudermannians of a real 0 have been tabulated from2 0 = 0 to 0 = 6. It is the object of this note to point out, however, that the preceding equations are not limited to real angles; but pertain to complex angles (9) throughout. That is, if i = V:=?, 13 = 1 + i12 = gd 0 = gd(0l + i02) complex cir. radians (10) and 0 = .0 + i02 = gd-'1 = gd-'(1, + i#2) complex hyp. radians (11) where ,1 and 12 are, respectively, real and imaginary components of 1; while 01 and 02 are the corresponding components of 0. Here 11 is essentially a real circular angle, and 01 a real hyperbolic angle; while i12, being an imaginary circular angle, may be regarded as a hyperbolic angle, and i02 being an imaginary hyperbolic angle, may be regarded as a circular angle. In this sense, both of the complex angles 13 and 0 may be considered as being composed of a circular and a hyperbolic component. Moreover, equations (7) and (8), when dealing wholly with the usual real quantities, may be rewritten in the forms: real cir. radians (12) I1 = gd 01 (12 = 02 = 0) (13) 01 = gd-ch1, (12 = 02 = 0) real hyp. radians

ENGINEERING: A. E. KENNELLY

840

PROC. N. A. S.

Methods of Solving Complex Gudermannian Equations.-It can be found, by trial, that any of the gudermannian equations (1) to (8), considered as extended to the complex plane, can be solved for any finite complex value of 0. Thus, let 0 = 0.5 + i 0.75 = 0.5 + i 1.1781 (14) where 02 is 1.1781 cir. radians or 0.75 quadrant, or 75 grades, or 670 30'. Then applying, for instance, equation (1), sin j = sin (i,% + i#2) = tanh (0.5 + i 1.1781)

which by Tables,3 is = 1.40579 + i 0.84585.

(15)

This equation may be solved by the use of the same Tables, or by Charts4 or by known fornulas. The solution is: =

+ i32

=

0.93737 + i 1.1547 cir. radians

(16)

where 0i = 0.93737 circular radians, and 32 = 1.1547 hyperbolic radians. This is the complex gudermannian of 0 = 0.5 + i 0.75 or 0.5 + i 1.1781. The same solution might be found by each and all of equations (1) to (6) taken as complex. The first set of complex gudermannians was obtained in this way, in a problem concerning electrical networks, which suggested the likelihood of such complex angles by physical analogy. There is. however, a shorter method of solution. lt may be shown that, given any pair of complex components 01 and 02, tan A

COS

02

(17)

and tanh ,82 = sin 2 cosh 0(

(18)

We may plot, on a f1132 plane, the points corresponding to successive pairs of 0102. It is evident from (17) and (18), that if we vary 02, keeping 01 constant, the values of 01 and 32 correspond to:

#1

=

tan-' (A sec 02)

(19)

and ,32 = tanh-1 (B sin 02)

(20)

where the constants are:

A = sinh 0, B = sech Oi.

(21) (22)

VOL. 14, 1928

ENGINEERING: A. E. KENNELLY

If, on the other hand, we vary 01, keeping 02 constant, = tan-' (C sinh 0E)

2= tanh-' (D sech 0A)

841

(23)

(24)

where the constants are:

C = sec 02 (25) D = sin 02. (26) Geometrically, equations (19) and (20) lead to a series of closed loops in the #1# plane, that may be called gudermannian loops of constant real component 01; while equations (23) and (24) lead to a series of curves which start from definite points on the j3s axis, and pass through the center of the 01 loops. These curves may be called gudermannian curves of constant imaginarv component 02. Principle of Cross-Substitution in Gudermannian Identities.-If we form the equations for antigudermannians or lambertians, we find: tanh 01 =

sin

cosh #2

(27)

and tan O,,

sinh cos

2

P

(28)

which compare with (17) and (18). In fact, we can form (27) and (28) out of (17) and (18), by mutually exchanging 01 and 2, and likewise 02 with #%. The same graph of gudermannians in the #1#2 plane may be converted into a graph of antigudermannians in the 0102 plane, with the aid of this cross substitution. There are a number of such cross-substitution formulas in the theory of gudermannians. For example, taking gd(0l + iA2) = 01 + i2 complex circular radians (29) and cross substituting as above, we have: gd(02 + i,31) = 0 + i0l complex circular radians (30) which is a formula of general application. Thus we have seen that

gd(0l + iA2)

=

gd(0.5 + il.1781)

=

0.93737 + il.1547

(31)

Cross substituting,

gd(l.1547 + i 0.93737) = 1.1781 + iO.5 (32) Again, if in (29), we assign the value zero to both 02 and i2, thus reducing (29) to (12), the ordinary real gudermannian equation, then (30) becomes: gd(O + i,1) = (O + i0l) (33)

842

ENGINEERING: A. E. KENNELLY

P-xoc. N. A. S.

or simply

gd(i,li) = ii = i gd-1f3

(34) Thus, if we take_from existing Tables of real gudermannians the relation gd 01 = gd 1.615 = (3l = 1.17814;

(35)

then

gd(il.17814) = il.615

(36)

or the gudermannian of an imaginary hyperbolic angle is its real antigudermannian taken as an imaginary. A Table of real gudermannians is therefore also a Table of imaginary gudermannians, when read reversely. In other words, existing Tables of real gudermannians serve also to evaluate imaginary gudermannians, by simple reversion. Equality of Tangent Products of Components in Complex Angles and Their Gudermannians.-It may be noted that if we multiply (17) by (18), or (27) by (28), we obtain tan 31.tanh (32 = tanh 01.tan 02 = tan y

(37)

which means that the tangent products are always equal. The identity also follows from cross substitution. The circular angle -y is that which the tangent to the constant-01 loops makes with the Y-axis, or the tangent to the constant-02 curves makes with the x-axis. In the same graph treated as an antigudermannian graph, these axes exchange. Practical Applications of Complex Gudermannians.-Complex gudermannians have various applications in physics and electrical engineering. Thus, if an infinite number of very long, equidistant, parallel, coplanar, straight wires are steadily maintained at successively equal and opposite potentials, the two dimensional field of potential-flux distribution in a plane perpendicular to the wires is a gudermannian complex, with equipotentials following loops of constant 01 and flux paths following curves of constant 02. This distribution has been cleverly worked out and graphed by Dr. Bernard Hague5 for the corresponding magnetic case in relation to dynamo-electric machines; but in algebraic terms only, and without reference to gudermannians or complex trigonometry. A similar set of curves has also been published, in relation to problems of electrical transmission lines, by R. S. Brown,6 without explicit gudermannian significance. The forms of complex gudermannian distribution graphs are therefore not new; although their trigonometrical meaning and implication may have escaped attention. Outline Table of Complex Gudermannians.-The accompanying Table gives values of 3,i and 32, to four decimal places, for various stepping values of 0A and 02. Formulas (1) to (6), or (17) and (18), enable, however, the computations to be carried to as many decimal places as existing

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CHEMISTRY: N. A. MILAS

PROC. N. A. S.

Tables of real circular and real hyperbolic functions will justify. It will been seen that for 0, = 0, ii% goes to i c. Beyond 01 = 2.5, the values of complex #, are confined to a small and nearly circular range, centered at 131 = 7r/2, 12 = 0. In the Table, 02 is given in three different alternative forms, in the first three columns, namely, in terms of 7r, of radians, and of quadrants, respectively. An ambiguity appears of ==nvr radians in the values of j3 and 02, where n is any integer; but ordinarily this does not involve uncertainty in the application of complex gudermannians. 1 "Theorie der Potenzial- oder cyklisch-hyperbolischen Functionen," C. Gudermann, Berlin, 1833. 2 "Smithsonian Mathematical Tables-Hyperbolic Functions," G. F. Becker and C. E. Van Orstrand, Washington, D. C., 1909. 3 "Tables of Complex Hyperbolic and Circular Functions," A. E. Kennelly, Harvard Univ. Press, 1914 and 1921. 4 "Chart Atlas of Complex Hyperbolic and Circular Functions," A. E. Kennelly, Harvard Univ. Press, 1914, 1921 and 1926. 6 "The Leakage Flux between Parallel Pole Faces of Circular Cross-Section," B. Drake, J. Inst. Elec. Engrs. London, Oct., 1923, 61, pp. 1072-1078. 6 "Use of the Tangent Chart for Solving Transmission-Line Problems," R. S. Brown, J. Am. Inst. Elec. Engrs., 1921, 40, p. 854.

NEW STUDIES IN POLYMERIZATION I. POLYMERIZATION OF STYRENE By NICHOLAS A. MILAs1 CmoIcAL LABORAIORY, PRINCrTON UNIVERSITY Communicated October 15, 1928

In the course of a series of expenments2 on the rates of auto-oxidation of iso-eugenol, iso-safrol, anethol, styrene, etc., the writer observed an abnormal decrease in the rate of oxygen absorption long before 10% of the substance in question had undergone oxidation. This decrease of the oxidation rate may be attributed to a rapid polymerization of styrene effected and accelerated by some product formed during the initial stages of the oxidation. In their studies of the polymerization of styrene to metastyrene under the influence of light and of heat by measuring the viscosity of styrene with time, Stobbe and Posnjak3 have arrived at the conclusion that the increase in the rate of polymerization of preparations that had been allowed to stand at room temperature for 14 days over those of freshly distilled styrene was unquestionably due to the formation in the former of an "Autokatalysator" or a "Polymerizationskern." In the same year Heinmann4 found that oxygen or ozone effected the polymerization of iso-