70 VECTOR ANALYSIS. 



where a', {?, y are the reciprocals of a, /3, y, and a, b, c, p, and q are 

 scalars, of which p is positive, will be most evident if we resolve it 

 into the factors 



aa + cosg{/3/3' + yy'} + smq{y/3'-/3y'}, 



of which the order is immaterial, and if we suppose the vector on 

 which we operate to be resolved into two factors, one parallel to a, 

 and the other in the /3-y plane. The effect of the first factor is to 

 multiply by a the component parallel to a, without affecting the other. 

 The effect of the second is to multiply by p the component in the 

 /3-y plane without affecting the other. The effect of the third is to 

 give the component in the /3-y plane the kind of elliptic rotation 

 described in No. 147. 



The effect of the same dyadic as a postf actor is of the same nature. 



The value of the dyadic is not affected by the substitution for a of 

 another vector having the same direction, nor by the substitution for 

 /3 and y of two other conjugate semi-diameters of the same or a 

 similar and similarly situated ellipse, and which follow one another 

 in the same angular direction. 



Def. Such dyadics we shall call cyclotonic. 



154. Cyclotonics which are reducible to the same form except 

 with respect to the values of a, p, and q are homologous. They are 

 multiplied by multiplying the values of a, and also those of p, and 

 adding those of q. Thus, the product of 



c^act' -f p 1 cos q l {/3/3' + yy'} + p l sin q l {yfi - /3y} 

 and a 2 aa' + p 2 cos q z {/3/3' + yy} + p z sin q 2 {y/3' - ,#y'} 



is a^aa' + p&t cos (q 1 + q 2 ) {/3fi' + yy} 



+PiPz sin (q l + q 2 ) {y/3' - #/}. 



A dyadic of this form, in which the value of q is not zero, or the 

 product of TT and a positive or negative integer, is homologous only 

 with such dyadics as are obtained by varying the values of a, p, and q. 



155. In general, any dyadic may be reduced to the form either of a 

 tonic or of a cyclotonic. (The exceptions are such as are made by 

 the limiting cases.) We may show this, and also indicate how the 

 reduction may be made, as follows. Let $ be any dyadic. We have 

 first to show that there is at least one direction of p for which 



$.p = ap. 

 This equation is equivalent to 



or, 



