602 



Mr. G. F. Becker on the Gudermannian 



the circle whose radius is the geometric mean of the axes, 

 this function has still other luminous properties. In a pure 

 shear the particles which in the unstrained state lie along a 

 radius at 45° to the axes lie after strain along the isocyclic 

 diameter, this material line having described an angle 

 tt/4 — JG(u). This same radius during strain sweeps over an 

 area u/2 which is an hyperbolic sector. In scission (slide, 

 shearing motion) the angle of rotation is 90° — Q(ii), and 

 one isocyclic diameter remains stationary while the other 

 sweeps an area u. The amount of shear is 2 cot Gt(u). 



Since all deformations can be resolved into pure shears 

 and scissions, all deformations can be reduced to terms of 

 the gudermannian complement, and no simpler treatment 

 has yet been suggested *. 



Consider an ellipsoid of three distinct axes hot, hj3, hy. 

 This may be regarded as the strain ellipsoid derived from a 

 sphere of radius (a-fty) 1 ^ by a dilatation of ratio //., and by 

 distortion. It is evidently legitimate to take this radius as 

 unity, so that <x/3y=l. The volume of the strain ellipsoid 

 is then f^A 3 , and for no strain A = l. Let a>/3>y and 

 consider the ratios of the axes, ha/h/3, h/3/hy, ha/hy. These 

 are all greater than unity if there is any strain at all. Hence 

 they may be represented by hyperbolic cosines, thus 



ha , lift i 7 ha. , 



— = cosh a: ^— = cosn o; y- = cosn c, 

 hS ha h<y 



hy 

 cosh a . cosh 6 = cosh c. 



and here 



If the gudermannian is substituted in this equation, 



cos gd a cos gd b— cos gd c, 



showing at once that the three angles are those subtending 

 the sides of a right-angled spherical triangle as shown in 



Fig. % 



fig. 2. The angles opposite gd a and gd b are marked respec- 

 tively v and </), but the complements of these angles are 



* See Smithsonian Math. Tables, introduction : and Bull. Geol. Soc. 

 kmer. vol. iv. 1893, p. 13. 



