84 VECTOK ANALYSIS. 



The constants a and /3 are to be determined by 



189. It will appear, on reference to Nos. 155-157, that every 

 complete dyadic may be expressed in one of three forms, viz., as a 

 square, as a square with the negative sign, or as a difference of squares 

 of two dyadics of which both the direct products are equal to zero. 

 It follows that every equation of the form 



d? =e - p ' 



where is any constant and complete dyadic, may be integrated by 

 the preceding formulae. 



NOTE ON BIVECTOE ANALYSIS * 



1. A vector is determined by three algebraic quantities. It often 

 occurs that the solution of the equations by which these are to be 

 determined gives imaginary values, i.e., instead of scalars we obtain 

 biscalars, or expressions of the form a + ib, where a and b are scalars, 

 and i = /s/ 1. It is most simple, and always allowable, to consider 

 the vector as determined by its components parallel to a normal 

 system of axes. In other words, a vector may be represented in the 



form m+yj+zk. 



Now if the vector is required to satisfy certain conditions, the solution 

 of the equations which determine the values of x, y, and 0, in the 

 most general case, will give results of the form 



n , 



* Thus far, in accordance with the purpose expressed in the footnote on page 17, we 

 have considered only real values of scalars and vectors. The object of this limitation 

 has been to present the subject in the most elementary manner. The limitation is 

 however often inconvenient, and does not allow the most symmetrical and complete 

 development of the subject in many important directions. Thus in Chapter V, and the 

 latter part of Chapter III, the exclusion of imaginary values has involved a considerable 

 sacrifice of simplicity both in the enunciation of theorems and in their demonstration. 

 The student will find an interesting and profitable exercise in working over this part of 

 the subject with the aid of imaginary values, especially in the discussion of the 

 imaginary roots of the cubic equation on page 71, and in the use of the formula 



in developing the properties of the sines, cosines, and exponentials of dyadics. 



