sponding points in the plane. It will involve no loss of generality to assume the radius of the sphere to be 1. Its equation then is

and OC being the vertical axis, P and Q corresponding points on the plane and sphere respectively, we have, by similar triangles,

From these and the equation of the sphere we readily obtain

and thence the values of , έ and 7 in terms of x and y, namely:

x2 + y2

x2 + y2 + 1

2x

x2 + y2 + 1

21

If it be desired

102. The Polar Transformation.

to present the formulæ of transformation in terms of tensor and amplitude, we may write

x=r cos 0, y = r sin 0,

ξ = cos cos 0, n= cos · sin 0, = sin &,

Thus the expression cos /(1 - sin) cis, in which and are independent of each other, suffices to represent all possible complex quantities.

By easy substitutions, &, n, are found, in terms of r and 0, to be

103. Agenda. Properties of the Stereographic Projection. (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 2iκπ, 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 w and w may be assumed to have the relation

W= w+a,

where a is a constant quantity (fixed in length and direction);* and the paths of B. 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 2¿к, 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 21π must lie in the direction OF, we conclude: in the w-plane be parallel and B in the z-plane will

If the paths of w。 and w straight lines, the paths of B

*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 2iKT.

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+aik,

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.