442 IX- DYNAMICS OF DEFORMABLE BODIES. 



equations 9) for r we have the complete geometrical representation 

 here applicable. Of the four quadrics the first is the most important. 

 The length r of the line OP is changed by the strain to r' which, 

 when the strain is small, differs from it by a small quantity, so that 

 the stretch 



~ r> \ r' r 



05) S r = -^ 



is a small quantity of the first order. But, since the angle between r 

 and r' is infinitesimal, we have to the first order, if q is the dis- 

 placement PP', 



56) r' = r -f gc 



-~ _ gcos(gr) _ 



r r 2 



Now if UjVjW are given by equations 50) the numerator becomes 



58) r*s r = s x x 2 + s y y* -f s z z* + 2^^ + 2^# + 2g,xy = #. 

 If we put this equal to unity we have 



59) ^ = ~ 2 > 



where r is the radius vector of the quadric 



60) z -l. 



This is called the elongation and compression quadric, and it is to be 

 noticed that the displacement of any of its points is in the direction 

 of the normal, for 



/w\ l 3% l d% 1 d% 



61) M = -^, #= *, ^ = * 



% ox % dy 2 cz 



Since any one of the six coefficients may be positive or negative, 

 the quadric may be an ellipsoid or an hyperboloid. In the latter 

 case not all the lines drawn from the origin will meet the surface, 

 and for those which do not r is imaginary and s r is negative. 



If we construct the conjugate hyperboloid, % = 1, those rays 

 which do not meet the first hyperboloid meet this, and the magnitude 

 of the compression is given by 



62) *<.. 



Lines that meet both hyperboloids at infinity and therefore have a 

 zero stretch or compression lie on the cone % = 0, asymptotic to the 

 two hyperboloids, and known as the cone of no elongation. 



All lines which are equally elongated with the stretch S, where 



63) S = i {s x x* + s y if + s,s* + 2g x yg + 2g y sx + 2g z xy], 

 lie on the cone 



