86 VECTOR ANALYSIS. 



With these definitions, a great part of the laws of vector analysis 

 may be applied at once to bivector expressions. But an equation 

 which is impossible in vector analysis may be possible in bivector 

 analysis, and in general the number of roots of an equation, or of the 

 values of a function, will be different according as we recognize, 

 or do not recognize, imaginary values. 



3. J)ef. Two bivectors, or two biscalars, are said to be conjugate, 

 when their real parts are the same, and their imaginary parts differ 

 in sign, and in sign only. 



Hence, the product of the conjugates of any number of bivectors 

 and biscalars is the conjugate of the product of the bivectors and 

 biscalars. This is true of any kind of product. 



The products of a vector and its conjugate are as follows : 



[jUL + lv] . [/X tv] = /X./U-f V. V 



[im + 1 v] X [ju tv] = 2 iv x AC 

 [M + iv] [M w] = {MM + vv} + i{vjui 



Hence, if JJL and iv represent the real and imaginary parts of a 

 bivector, the values of 



fA.fJL + V.V, JUiXV, fJLjUL + VV, V/JL JULV, 



are not affected by multiplying the bivector by a biscalar of the form 

 ib, in which a 2 +6 2 = l, say a cyclic scalar. Thus, if we set 



we shall have 



fjf iv = (a i b) [//. iv], 

 and 



[X + 1 v] . \JJL iv ] = [/* + tv] . [fj. /]. 

 That is, 



and so in the other cases. 



4. Def. In biscalar analysis, the product of a biscalar and its conju- 

 gate is a positive scalar. The positive square root of this scalar is 

 called the modulus of the biscalar. In bivector analysis, the direct 

 product of a bivector and its conjugate is, as seen above, a positive 

 scalar. The positive square root of this scalar may be called the 

 modulus of the bivector. When this modulus vanishes, the bivector 

 vanishes, and only in this case. If the bivector is multiplied by a 

 biscalar, its modulus is multiplied by the modulus of the biscalar. 

 The conjugate of a (real) vector is the vector itself, and the modulus 

 of the vector is the same as its magnitude. 



5. Def. If between two vectors, a and (5, there subsists a relation 

 of the form 



