62 VECTOR ANALYSIS. 



Now the supposed property of the direction of i requires that when p 

 coincides with i and dp is perpendicular to i, dp shall be perpendicular 

 to p', which will then be parallel to a. But if dp is parallel to j or k, 

 it will be perpendicular to i, and dp will be parallel to /3 or y, as the 

 case may be. Therefore /3 and y are perpendicular to a. In the same 

 way it may be shown that the condition relative to j requires that y 

 shall be perpendicular to /3. We may therefore set 



<1> = ai'i 4- bj'j 4- ck'k, 



V V 



where i', j', k', like i, j, k, constitute a normal system of unit vectors 

 (see No. 11), and a, b, c are scalars which may be either positive or 

 negative. 



It makes an important difference whether the number of these 

 scalars which are negative is even or odd. If two are negative, say a 

 and b, we may make them positive by reversing the directions of i' 

 and j'. The vectors i', j', k' will still constitute a normal system. 

 But if we should reverse the directions of an odd number of these 

 vectors, they would cease to constitute a normal system, and to be 

 superposable upon the system i, j, k. We may, however, always set 



either 3> = ai'i + tyj+cVk, 



or $ = { ai'i + bj'j 4- ck'k } , 



with positive values of a, b, and c. At the limit between these cases 



are the planar dyadics, in which one of the three terms vanishes, and 



the dyadic reduces to the form 



ai'i + bj'j, 



in which a and b may always be made positive by giving the proper 

 directions to i' and /. 



If the numerical values of a, b, c are all unequal, there will be only 

 one way in which the value of <3? may be thus expressed. If they are 

 not all unequal, there will be an infinite number of ways in which 3? 

 may be thus expressed, in all of which the three scalar coefficients 

 will have the same values with exception of the changes of signs 

 mentioned above. If the three values are numerically identical, we 

 may give to either system of normal vectors an arbitrary position. 



136. It follows that any self -conjugate dyadic may be expressed 



in the form . , T , 7 7 



an 4- fyj 4- c/ck, 



where i, j, k are a normal system of unit vectors, and a, b, c are 

 positive or negative scalars. 



137. Any dyadic may be divided into two parts, of which one shall 

 be self -conjugate, and the other of the form Ixa. These parts are 

 found by taking half the sum and half the difference of the dyadic 

 and its conjugate. It is evident that 



