> 166] PURE STRAIN. 



431 



By means of either of the quadrics 9 and <p ' we may thus 

 obtain the mutual directional relations of every pair of vectors 

 OP, OP' drawn from the origin, the relation being that of radius 

 vector and normal at a given point, and reciprocal with respect to 

 the two quadrics. 



We see that in the application to strain any point P lying on 

 the quadric cp = E 2 is displaced to a point P' on the quadric 

 (p 1 = + E 2 . 



If we call the feet of the perpendiculars from the origin on the 

 tangent planes at P and P' respectively Q and Q' and write p and p' 

 for the lengths OQ and OQ' we have 



p = r cos K) = 

 13) 



p' = r' cos (rr') = 

 so that 



14) pr' = p'r = xx' -f yy + e 



The quadrics (p and V are then said to be reciprocal to each 

 other with respect to a sphere of radius E. Since for the axes of 

 the quadric p = r and/ = r', we have the relation between the axes, 



15) rr' = E 2 , 



so that the axes of reciprocal quadrics are inversely proportional. 



Since all lines in the same direction are stretched in the same 

 ratio a line OS of length p = S in the direction of OP is strained 

 into a line OS' of length $' in the direction OP\ so that 



Now if a, /3, <y and cc', /3', y' are the direction cosines of OP and 

 OP' respectively, equations 9) and 12) may be written, when divided 

 through by r and r' respectively, 



L a t = aa + ftp + g^ 



