Complement and Imaginary Geometry. G03 



more interesting than the angles themselves. Putting 

 A = 90°-<£ and B = 90°— v, it appears at once that 



tanh a cos G(a) 



tan A = 

 tanB = 



sinh b cut (x(6) ' 



tanh b cosG(^) 

 sinh a ~~ cot Q(a) ' 



Here it is known that A is the angle which the circular 

 section of the strain ellipsoid under discussion makes with 

 its greatest axis *, and it is evident that B may be con- 

 sidered as the angle made by the circular section of a 

 second ellipsoid with its greatest axis. The axes of this 

 second ellipsoid are 1/ha, l/h/3, 1/hy ; in other words, it is 

 the reciprocal of the first. A and B are each in general 

 less than 7r/4, a value which they reach only for vanishing 

 strain. 



A very close relation exists between a strain ellipsoid and 

 its reciprocal. This is most readily seen in the case of a 

 mass subjected only to finite, homogeneous deformation, 

 which can always be represented by two shears of ratio a. 

 and j3. If these two shears have their axes of extension in 

 common they deform the sphere into an ellipsoid whose 

 axes are «/3, 1/a, lj/3. If the same two shears are com- 

 bined by their contractile axes, they yield an ellipsoid 

 whose axes are ljaft, a, and /3. This second ellipsoid is 

 thus the reciprocal of the first. The loads (or initial stresses, 

 or stresses into areas) are the same in absolute value in 

 the two cases, but with signs reversed ; so that equal finite 

 loads of opposite signs produce deformations of reciprocal 

 ratios. 



A shear results from the action of two loads, Q and P, at 

 right angles to one another if P= — Q. Q acting by itself 

 would produce a dilatation of ratio say li u and P a cubical 

 contraction of ratio say l//> 2 . Acting together to produce a 

 shear they must give a dilatation h=h i /h 2 . But since a 

 shear is undilatational, h = 1. Therefore equal forces of oppo- 

 site signs produce not merely deformations but strains of 

 reciprocal ratios. 



Hence an elastic, homogeneous sphere subjected to a finite 

 homogeneous strain and then allowed to vibrate is converted 

 into the reciprocal ellipsoid at the opposite phase. This 



* Smithsonian Math. Tables, p. xxxii. 



