VECTOR ANALYSIS. 69 



Def. The displacement represented by the equation 



P=-~P 



is called inversion. The most general case of a homogeneous strain 

 may therefore be produced by a pure strain and a rotation with or 

 without inversion. 



150. If 



< = ai'i -f bj'j + ck'k, 



$.$o = a>W + &*// + c* k'k', 

 and 3> & = a?ii + b'*jj+c 2 kk. 



The general problem of the determination of the principal ratios and 

 axes of strain for a 'given dyadic may thus be reduced to the case of 

 a right tensor. 



151. Def. The effect of a pref actor of the form 



aaa + 6/3/3' + cyy', 



where a, b, c are positive or negative scalars, a, ft, y non-complanar 

 vectors, and a, /3', y their reciprocals, is to change a into act, /3 into 

 b/3, and y into cy. As a postfactor, the same dyadic will change a' 

 into act', ft' into b/3', and y' into cy. Dyadics which can be reduced to 

 this form we shall call tonic (Gr. reivw). The right tensor already 

 described constitutes a particular case, distinguished by perpendicular 

 axes and positive values of the coefficients a, b, c. 



The value of the dyadic is evidently not affected by substituting 

 vectors of different lengths but the same or opposite directions for 

 a> ft, y> with the necessary changes in the values of a, ft, y, defined 

 as reciprocals of a, ft, y. But, except this change, if a, b, c are 

 unequal, the dyadic can be expressed only in one way in the above 

 form. If, however, two of these coefficients are equal, say a and 6, 

 any two non-collinear vectors in the a- ft plane may be substituted 

 for a and ft, or, if the three coefficients are equal, any three non- 

 complanar vectors may be substituted for a, ft, y. 



152. Tonics having the same axes (determined by the directions 

 of a, ft, y) are homologous, and their multiplication is effected by 

 multiplying their coefficients. Thus, 



' + C 2 yy'} 



Hence, division of such dyadics is effected by division of their co- 

 efficients. A tonic of which the three coefficients a, b, c are unequal, 

 is homologous only with such dyadics as can be obtained by varying 

 the coefficients. 



153. The effect of a pref actor of the form 



or aaa + pcosq{ftft' + yy'} -f psinq{yft'-fty'}, 



