VECTOR ANALYSIS. 63 



Now %{3?+3> c ] is self-conjugate, and 



(See No. 131.) 



Rotations and Strains. 



138. To illustrate the use of dyadics as operators, let us suppose 

 that a body receives such a displacement that 



p' = $.p, 



p and p being the position- vectors of the same point of the body in 

 its initial and subsequent positions. The same relation will hold of 

 the vectors which unite any two points of the body in their initial 

 and subsequent positions. For if p lt p 2 are the original position- 

 vectors of the points, and /o/, p% their final position-vectors, we have 



Pl=$'Pl> PZ=&P2> 



whence , , , r . 



Pz-pi ==< -U>2-/>iJ- 



In the most general case, the body is said to receive a homogeneous 

 strain. In special cases, the displacement reduces to a rotation. 

 Lines in the body initially straight and parallel will be straight and 

 parallel after the displacement, and surfaces initially plane and 

 parallel will be plane and parallel after the displacement. 



139. The vectors (a; a-') which represent any plane surface in the 

 body in its initial and final positions will be linear functions of each 

 other. (This will appear, if we consider the four sides of a tetra- 

 hedron in the body.) To find the relation of the dyadics which 

 express a-' as a function of or, and p as a function of />, let 



Then, if we write X', /*', v for the reciprocals of X, /x, i/, the vectors 

 X', /A', v become by the strain a, /3, y. Therefore the surfaces //Xi/, 

 i/xX', X'XM' become /3xy, yxa, ax/3. But /x'xi/, i/xX', X'Xju' are the 

 reciprocals of /xXi>, i/xX, Xx/u. The relation sought is therefore 



(/ = {/3xy v.Xv -f yxa i/xX + ax/3 Xx/*}.(r. 



140. The volume X'./x'xi/ becomes by the strain a./Sxy. The 

 unit of volume becomes therefore (a./3xy)(X.ywXi/). 



Def. It follows that the scalar product of the three antecedents 

 multiplied by the scalar product of the three consequents of a dyadic 

 expressed as a trinomial is independent of the particular form in 

 which the dyadic is thus expressed. This quantity is the determinant 

 of the coefficients of the nine terms of the form 



aii + bij + etc., 



