103. Agenda. Properties of the Stereographic Pro

jection.

(1).

Prove analytically that a circle, or a straight line in the plane, corresponds to a circle on the sphere.

(2). Prove geometrically that any two lines in the plane cross each other at the same angle as the corresponding lines in the sphere. (Cf. Art. 107.)

(3). Show that to the centre of projection correspond all points at infinity in the plane, and that it is therefore consistent to say: there is in the complex plane but one point at infinity.

(4). Show that meridians on the sphere through the centre of projection, and parallel horizontal circles on the sphere correspond respectively to straight lines through the origin and concentric circles in the plane.

XVII. PLANAR ORTHOMORPHOSIS.

104. W-plane and Z-plane. In the graphical representation of an equation connecting two complex varying quantities w and z, it conduces to clearness of delineation and exposition to separate the figures representing the variations of w and z, and to speak of the w-plane and the z-plane as though they were distinct from one another. This language and procedure help us to see more clearly that the plane with the w-markings upon it has a distinctive character and presents in general an appearance different from that which it has when its markings represent the variations of the functions of w, and to distinguish more easily the two groups of markings from one another.

It is the purpose of the present section to describe the planar orthomorphosis of some of the functions that have been defined in the foregoing pages, that is, to cause the

point Q, the affix of w, to traverse the w-plane in a specified manner, and to mark out the paths that P, the affix of a function of w, will follow, in consequence of the assumed variations of w.

105. The Logarithmic Spirals of Bw Non-intersecting. The function B, is singly periodic; that is, there is only one quantity, the period 2¿‹π, multiples of which, when substituted for w, will render Bw 1, (Art. 81). If now and Q, the affixes of wo and w, move in the w-plane upon parallel straight lines, the variable quantities W0 and w may be assumed to have the relation +a,

w=w

where a is a constant quantity (fixed in length and direction); and the paths of Bw. and Bw will either not intersect at all, or will coincide throughout their whole extent. For, in order that the two paths may have a point in common there must be a pair of values w。, w。+a, for which

Bw. + a = Bwo,

and a must be a multiple of 2iкπ (Art. 81). But if a be a multiple of 2iкя, then for all values of w

and the two w-curves have all their points common. Hence, since in the construction of Fig. 36, Art. 68, a vector representing 2¿π must lie in the direction OF, we conclude:

If the paths of w。 and w in the w-plane be parallel straight lines, the paths of Bw and Bw in the z-plane will

*This is merely a way of saying, that if a link, or rod, while remaining parallel to a fixed direction, move with one of its extremities upon a fixed straight line, its other extremity generates a second straight line parallel to the first.

be coincident, or distinct and not intersecting, according as the intercept made by the two w-lincs on the modular normal is or is not a multiple of the period 2iKm.

106. Orthomorphosis of Bw. The fixed elements in the w-plane are the real axis, the modular line and the modular normal,-in Fig. 36, the lines OJ, ET and OF; in the z-plane they are the real axis and the unit circle.

By the operation of exponentiation, indicated by Bw, a straight line in the w-plane is transformed, metamorphosed, into a logarithmic spiral (Art. 80). Hence if the variable elements of the w-plane be assumed to be straight lines, in the z-plane they will be logarithmic spirals.

Assigning as the path of wo a straight line OS passing through the origin (Fig. 40), write

w=w+aix,

in which a is a real quantity. The path of w, for a given value of a, will then be a line EC, parallel to OS, to which will correspond in the z-plane, a logarithmic spiral E'C' (Fig 41). In particular to the path of w, corresponds the spiral z, that passes through the intersection of the real axis with the unit circle.

To the straight lines in the w-plane, obtained by giving different values to a, there correspond in the z-plane, so long as a is less than 27 and not less than o, distinct nonintersecting logarithmic spirals (Art. 105). And since OC=c= =ma (Art. 72), when a varies from o to 27, C varies from o to 2m, or as represented in Fig. 40, from o to OC, and when EC moves from the position OS to the position E,S,, the corresponding logarithmic spiral makes a complete revolution and sweeps over the entire z-plane.

To every point in the z-plane corresponds one and only one point in the band between the parallels OS。, ES, whose width is 2mπ cos (— ẞ), and also one and only one point within every band, in the w-plane, having this width and parallel to OS. The construction therefore shows graphically how, to every value of z, there correspond an infinite number of values of w, namely all the values w+2kiкя, wherein k is any integer. The successive affixes of w, w+2iKπ, w2iKπ, w+4iкπ, etc., are obviously situated at the division-points of equidistant intervals, each equal to 2mπ, along a straight line through the affix of w parallel to OF.

Whenever a w-line crosses OF, the corresponding z-spiral crosses the unit circle (Art. 68). Hence, to points in the w-plane below or above the modular normal OF, correspond respectively points in the z-plane within or or without the unit circle. Thus the shaded and unshaded portions of Figs. 40, 41 correspond respectively.

The points where any spiral in the z-plane crosses the real axis are those for which Bw is real. But the necessary and sufficient condition for the vanishing of the imaginary term of Bw is

sin

v cos Businß

O,

(Art. 73.)

m

wherein is any integer; and the locus of the equation last written is a straight line parallel to OT and distant from the origin a multiple of mπ. Hence, whenever a w-path crosses such a line the corresponding z-spiral crosses the real axis. (Cf. Arts. 72, 80.)

As particular constructions we have: When the w-lines are parallel to the modular line OT, the z-spirals degenerate into straight lines passing through the origin; and when the w-lines are parallel to the modular normal OF, the z-spirals become circles concentric with the unit circle. (Cf. Art. 71.)

107. Isogonal Relationship. Any two spiral paths of Bw cross one another at the same angle as the corresponding straight paths of w.

Let 。, Þ。, v。, u。 and 0, p, v', u' be two sets of corresponding values of 0, p, v' and u' (Fig. 36). It was shown in Art. 72 that mo

=

v'; hence

v′ — v。 = (u′ — u ̧) tan (Þ — ß),

whence, since p=b"' and p。=bu。 (Art. 73),

Þ2 (0-0) -
0。) __Þu. u'-u

כי

p-po m bibl。

tan ( — ẞ);