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, In this product, tensor of a XB-a X tensor of ẞ by definition, and if tensor of B1, the complex quantity a × ẞ appears as the product of its tensor and a unit factor ẞ, a complex unit, which when applied as a multiplier to real quantity a, does not change its length but turns it out of the real axis into the direction of B. axb α Fig. 31. 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 var 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, 0 the amplitude of the complex Ө =cos B=OM+iX MB= cos 0+ i sin 0. As an abbreviation for cos + 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 (pròduct of complex quantities, as above defined) is expressed in the formula, (a cis ) X (bcis y) = ab cis (+). 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 y = – 4, it becomes (a · cis $) × (a · cis [— $]) = a X a· cis o = a2. a cis a cis(-) Fig. 33. 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; that is, by virtue of the definition of Art. 6, cis & and cis (—) are reciprocal to each other and we may write In like manner, since it is now evident that (a cis p) X (/a/cis p) = a/ acis o = 1 If now in the formula expressing the law of multiplication we write band for b and the law of geometric division, 62. prove: respectively, we have, as (a cis p)/(b cis) a/b⋅ cis (—). = Agenda. Properties of cis . If n be an integer (1). (a cis $)" : an · cis n o. (2). cis (± 2 n ) = cis . (3). cis ([2 n + 1] π) = — cis ø. (4). cis (+ [2 n±}]π) = ± i cis &. (5). cis (+ [2 n ± } ] π) == i cis &. (6). Show that the ratio of two complex quantities having the same amplitude, or amplitudes that differ by 2π, is a real quantity. (7). Show that the ratio of two complex quantities having amplitudes that differ by ± is a purely imaginary quantity. 63. The Imaginary Unit. By definition (iii) of Art. 57 cis π/2i is an imaginary having a unit tensor; it is therefore called the imaginary unit. Its integral powers form a closed cycle of values; thus: and the higher powers of i repeat these values in succession; that is, if n be an integer, ¿4n+1=i, 24n+2= I, 24n+3= ·i, ¿4n+4=1, and these are the only values the integral powers of i can acquire. 64. The Associative and Commutative Laws. Let a, b, c be the tensors, 4, 4, x the amplitudes of a, ß, y respectively; that is, a=a* cis &, ẞ=b⋅cis 4, y=c*cis x. Then, by the law of geometric multiplication, a × (ẞ × y) = (a · cis ¢) × ([b · cis 4] × [c cis x]) =ax (b× c) · cis (☀ + [¥ +x]); and by the same process, (a XB) Xy=(a × b) × c • cis ([☀+¥]+x). But, by the rules of algebraic multiplication and addition, ax (bc)=(ab) X c and +(y+x)=($+4)+x; ... ax (BXy)=(a X B) Xy, which is the associative law for multiplication. And again, by the law of geometric multiplication, and similarly ẞXa=ba cis (¥ X $). But, by the rules of algebraic multiplication and addition, ax b = b xa and +4=4+; ... axẞ=ßxa, which is the commutative law for multiplication. The letters here involved may obviously represent either direct factors or reciprocals, and the sign X may be replaced at pleasure by the sign/, without affecting the proof here given (see Art. 61). Hence, as in Arts. 32, 33, with real magnitudes, so with complex quantities the associative and commutative laws for multiplication and division have their full expression in the formulæ, (Xax B) = ( a) × (ß), 65. The Distributive Law. From the definitions of geometric addition and multiplication (Arts. 58, 60) the law of distribution for complex quantities is an easy conse quence. The constructions of the subjoined figure, in which a, ß and mcis represent complex quantities will B bring this law in direct evidence. The operation m cis changes OA into OA', AB= A'Finto AE= A'B', and OB into OB', that is, turns each side of the triangle OAB through the angle AOA' and changes its length in the ratio of m to I, producing the similar triangle OA'B', in which OA'mcis Xa, A'B' =mcis @ X ß, OB'm' cis X (a+ẞ). But, by the rule of geometric addition, |