68 VECTOR ANALYSIS. 



I, or at least capable of expression with any required degree of 

 accuracy as a root of I. 



It should be observed that the value of the above dyadic will not 

 be altered by the substitution for a of any other parallel vector, or 

 for /3 and y of any other conjugate semi-diameters (which succeed one 

 another in the same angular direction) of the same or any similar and 

 similarly situated ellipse, with the changes which these substitutions 

 require in the values of a', ft, y. Or, to consider the same changes 

 from another point of view, the value of the dyadic will not be altered 

 by the substitution for a of any other parallel vector or for /3' and y 

 of any other conjugate semi-diameters (which succeed one another in 

 the same angular direction) of the same or any similar and similarly 

 situated ellipse, with the changes which these substitutions require in 

 the values of a, /3, and y, defined as reciprocals of a, /&, y. 



148. The strain represented by the equation 



p = {aii + bjj + ckk} . p 



where a, b, c are positive scalars, may be described as consisting of 

 three elongations (or contractions) parallel to the axes i, j, k, which 

 are called the principal axes of the strain, and which have the pro- 

 perty that their directions are not affected by the strain. The scalars 

 a, b, c are called the principal ratios of elongation. (When one of 

 these is less than unity, it represents a contraction.) The order of 

 the three elongations is immaterial, since the original dyadic is equal 

 to the product of the three dyadics 



aii +jj -f- kk, ii + bjj + kk, ii +jj -f ckk 



taken in any order. 



Def. A dyadic which is reducible to this form we shall call a right 

 tensor. The displacement represented by a right tensor is called a 

 pure strain. A right tensor is evidently self -conjugate. 



149. We have seen (No. 135) that every dyadic may be expressed 



in the form 



(ai'i + bj'j + ck'k], 



where a, b, c are positive scalars. This is equivalent to 



and to 



{i'i +j'j + Vty .{aii + bjj + ckk}. 



Hence every dyadic may be expressed as the product of a versor and 

 a right tensor with the scalar factor 1. The versor may precede or 

 follow. It will be the same versor in either case, and the ratios of 

 elongation will be the same; but the position of the principal axes 

 of the tensor will differ in the two cases, either system being derived 

 from the other by multiplication by the versor. 



