The student can easily verify these results by making the transformations independently.

The hyperbolic forms, obtained from the foregoing by putting <= I are important, and are of frequent application in the integral calculus. They are given in Art. 56. As there explained, for real values of u the positive sign before the radicals must then be chosen. The forms for the inverse circular functions, got by putting xi, are less frequently useful.

98. Agenda. Prove the following, n being an integer:

(2). tan ̧1× (≈2 − 1) / (~2 + 1) =log ̧2+2ni«T.

(4). cos(x + y) = m (cos ẞ cosh-'p- sin ẞ cos'σ) +im (sin ẞ cosh-p+ cos ẞ cos'σ),

where

p=±√(s+ √s2−x2), o=±√ ( s − √ 52 — x2), s = √(x2 + y2+1), and κm cis ß.

=m

(5). sin(x+y)= m (cos ẞ cosh-1P- sin ẞ sin1) +im (sin ẞ cosh-P+ cos 3 sin-'),

where, as before, κ=m cis ß, and

P=±√(S+√ S* − X2), Σ=±√(S−√ S2 — X2),

S= { (X2 + Y2 + 1 ) = { (x2 + y2 + m2) ! m2,

X=(-xsin ẞ+y cos 3) / m, Y= (x cos ẞ+y sin ẞ)/m. (6). Reduce tang' (x + y) to the form u + iv.

CHAPTER V.

GRAPHICAL TRANSFORMATIONS.

XVI.

ORTHOMORPHOSIS UPON THE SPHERE.

99. Affix, Correspondence, Morphosis. Every complex quantity, defined geometrically by a vector drawn from the origin with proper length and direction, determines uniquely a point in its plane, namely the extremity of the vector; and conversely, to every point in the plane corresponds one and only one complex quantity. It is convenient therefore to assign, as the geometrical representative, or affix,* of a given complex quantity, a point in a plane, to note its different states by a series of affixes, and to represent a continuous change in it by a line, its path, in general not straight.

This relation between point and complex quantity is described as a one-to-one correspondence, and the spreading out upon the plane of the points or paths of a varying complex quantity is its morphosis in the plane, or its planar morphosis.

100. Stereographic Projection. By means of a stereographic projection we may establish a one-to-one correspondence between the plane and a sphere, so that a point upon the plane determines uniquely a point on the sphere, and vice versa; and the spreading out of all the

* C. Jordan: Cours d'Analyse de l'École Polytechnique, Vol. I, p. 106, Art. 116.

points or lines on the sphere that correspond to the affixes or paths of a complex quantity in the plane, will be its morphosis, more specifically its orthomorphosis* upon the sphere. We accomplish the transformation in the following

manner.

We place the sphere with its center at the origin of complex quantities in the plane. Regarding the plane as fixed in a horizontal position, all projecting lines are drawn from the upper extremity of the vertical diameter as a center of projection (Fig. 39 of Art. 101). Then if from this center of projection a straight line be drawn to any point in the plane, it will cut the spherical surface in one other point. The two points thus uniquely determined, one in the plane, the other on the sphere, are said to correspond to each other, or to be corresponding points. Thus a complex quantity may have an affix in the plane and a corresponding affix on the sphere.

Points in the plane inside the great circle in which it cuts the sphere correspond to points on the lower hemisphere; points in the plane outside this circle have their correspondents on the upper hemisphere.

ΙΟΙ.

Transformation Formulæ.** In order to connect algebraically the two forms of the complex quantity, in the plane and on the sphere, assume as the origin of co-ordinates in both systems the center of the sphere, let έ and be the horizontal, the vertical co-ordinates of points on the sphere, x and y the co-ordinates of corre

* An orthomorphic transformation of the plane into the spherical surface. This kind of transformation is called orthomorphosis by Cayley: Journal für die reine und angewandte Mathematik, Bd. 107 (1891), p. 262, and Quarterly Journal of Mathematics, Vol. XXV (1891), p. 203. See also the first footnote to Art. 108.

**Ct. Klein: Vorlesungen über das Ikosaeder, p. 32.

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

η

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

If it be desired

102. The Polar Transformation.

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

and are independent of each other, suffices to represent all possible complex quantities.

By easy substitutions, έ, n, are found, in terms of r