the category of algebraic quantity, a new kind of quantity must therefore be defined, or more properly, a new definition of algebraic quantity in general must be given.

Having assigned some fixed direction as that in which all real quantities are to be taken, we adopt a straight line having this direction as a line of reference, call it the real axis, and determine the directions of all other straight lines in the plane by the angles they make with this fixed one. Line-segments having directions other than that of the real axis are the new magnitudes that now demand consideration. They are called vectors. They have two determining elements length and the angle they make with the real axis.

(i). Its length, taken positively, is called the tensor of the magnitude, and the arc-ratio of the angle it makes with the real axis is called its amplitude (or argument)..

Classified and defined with respect to amplitude, the magnitudes themselves are:

(ii). Real, if the amplitude be o or a multiple of π ; (iii). Imaginary, if the amplitude be π/2 or an odd multiple of 2.

(iv).

π

Complex, for all other values of the amplitude.

In general, therefore, vectors in the plane represent complex quantities, but in particular, when parallel to the real axis they represent real quantities; when perpendicular to it, imaginary.

Any quantity is by definition uniquely determined by its tensor and amplitude, and hence:

(v). Two quantities are equal if their tensors and their amplitudes are respectively equal, the geometrical rendering of which is two magnitudes, or vectors, are equal if (and only if) they are at once parallel, of the same sense, and of equal lengths.

The algebra of complex quantities, like that of real quantities, is developed from the definitions of the fundamental algebraic operations: addition and subtraction, multiplication and division, exponentiation and the taking of logarithms. These operations applied to magnitudes represented by straight lines in the plane are called algebraic by reason of their identity with those of the analysis of real quantities, but specifically geometric, because each individual operation has its own unique geometrical configuration. On the other hand, the algebraic processes applied to real quantities may be described as geometric addition, multiplication, involution, etc., in a straight line.

58. Geometric Addition. Regarding lines for our present purpose as generated by a moving point, the operation of addition is defined to mean that a point P, free to move in any direction, is successively transferred forwards or backwards, that is, in the positive or negative sense as marked by the signs and —, through certain distances designated by appropriate symbols a, B, y.. Thus the sum+a - B+y, in which a, ß, y represent vectors in the plane (or in space), joined to form a zig-zag, as shown in the accompanying figure, may be read off as follows, the arrow-heads indicating direction of motion forwards: Move forwards through distance a to A, then backwards through distance B to B', then forwards through distance y to C', and the result is the same as if the motion had taken place in a direct line from O to C'; this fact is expressed in the equation

&

Β'

α

୪

B

A

Fig. 29.

B

C'

If not already contiguous, the magnitudes that form the terms of a sum, by changing the positions of such as require it without changing their direction, may be so placed that all the intermediate extremities are conterminous. Geometric addition may therefore be defined as follows:

The sum of two or more magnitudes, placed for the purpose of addition so as to form a continuous zig-zag, is the single magnitude that extends from the initial to the terminal extremity of the zig-zag.

59. The Associative and Commutative Laws for geometric addition in the plane are deduced as immediate consequences of its definition. For in the first place, the

ultimate effect is the same whether a transference is made

Fig. 30.

and in the second place, by (v) of

Art. 57, in ABCD, a parallelogram, AD = BC=Y, DC=AB=B, and by the definition of addition AB+ BC =AC=AD+ DC, whence

B+y=y+B.

One or more of the terms may be negative. Expressing this fact by writing ±a, ± ß, ±y in place of a, ß, y, the two resultant equations of the last paragraph become

(±a ± ß) ± y=±a+(±ß ±y) =

±y±ß= ±ß ±y,

=

which express the associative and commutative laws for addition and subtraction.

It is evident that the principles of geometric addition apply equally to vectors in space, or in the plane, or to segments of one straight line. In particular algebraic addition (Art. 3) may be described as geometric addition in a straight line.

60. Geometric Multiplication. The geometric product of two magnitudes a, ẞ, is defined as a third magnitude y, whose tensor is the algebraic product of the tensors of the factors and whose amplitude is the algebraic sum of their amplitudes, constructed by the rules for algebraic product and sum. (Arts. 5, 3.)

If one of the factors be real and positive, the amplitude of the other reappears unchanged as the amplitude of the product, which is then constructed, by the algebraic rule, upon the straight line that represents the direction of the complex factor, and it was proved in Art. 18 that in such a construction an interchange of factors does not change the result. Hence, if a be real and positive and ẞ complex,

αχ β=β χα.

In this product, tensor of a X ß= =a X tensor of ẞ by definition, and if tensor of ẞ=1, the complex quantity aẞ appears as the product of its tensor and a unit factor ẞ, a complex unit, which when applied as a multiplier to a real quantity a, does not change its length but turns it out

αχ β

α

Fig. 31.

of the real axis into the direction of ß. Any such complex unit is called a versor. Let tsr stand for tensor, vsr for versor; then every complex quantity a can be expressed in the form.

α= tsr a X vsr a.

This versor factor is wholly determined by its amplitude, in terms of which it is frequently useful to express it. For this purpose let i be the versor whose amplitude is π/2, Ө the amplitude of the complex

unit B, OX the real axis, BM the perpendicular to OX from the terminal extremity of B. Then MB sin 0, OM= cos 0,

=

and by the rule of geometric addition.

B=OM+i × MB = cos

B

As an abbreviation for cos 0 + i sin 0 it is convenient to use cis 0, which may be read: sector of 0. In this symbolism, the law of geometric multiplication (product of complex quantities, as above defined) is expressed in the formula,

(a · cis $) × (b · cis ¥) = a × b • cis (Þ ·+ 4).

It is obvious that algebraic multiplication, described in Art. 5, is a particular form of geometric multiplication, being geometric multiplication in a straight line.

61. Conjugate and Reciprocal. If in the last equation b =a and 4: - p, it becomes

(a · cis $) × (a · cis [— $]) = a × a·cis o = a2.

The factors of this product, a cis and a cis (-), are said to be conjugate to one another, and we have the rule:

The product of two conjugate complex quantities is equal to the square of their tensor.

If this tensor be 1, the product reduces to

cis · cis (— 4) = 1 ;