Complement and Imaginary Geometry . 607 



in terms of the other functions*. Now 



£■ = cosh b = sin (2B ) _ sin G(a) 

 P «7 ~ cosh a ~~ sin (2A) ~" sin G(b) ' 



and thus the angle B is a significant feature of the strain 

 ellipsoid itself, as well as of the reciprocal ellipsoid. 



It is noteworthy that in the simple case of pure deforma- 

 tion discussed the circular functions give the relative positions 

 of points in the mass to one another, while the hyperbolic 

 functions appertain to the lines of flow or to the absolute 

 motions of the particles. This distinction is not preserved, 

 however, in the case of rotational strains. 



Having thus outlined the parts played in finite homo- 

 geneous strain, flow, and rupture of rocks and other solids 

 by the gudermannian complement and the angles A and B, 

 it may be well to write down for reference a few of the 

 formulae connecting these functions : — 



. . _ cot G{a) sinh a 

 cot G{c)~ sinh c' 



A cos G(b) tanh b 

 cos A— - 



cot A = 



cos G(c) tanh c 



sin B . 



• n , x = sin B c 

 sin G(a) 



cot G(5) sinh b 



. sin B . 



cos A= - — ~, . = sin B cosL a. 

 Sill G(a) 



cos G(a) tanh a* 



-r, cos G(a) tanh a 



cos B = ~^ = - r— . 



cos \jt(c) tanh c 



COS B = — — TTTTx = SID. A COsll 6. 



sin G(6) 



. ,. cot G(b) sinh b 



sin B = .,; ' = -^—, — . 



cot G{c) sinh c 



, -r> cot G (a) sinh a 



cot B = 7 ^-( ~ — — - . 



cos Gib) tanno 



tan A and B= sin G(c)= sin G(a) sin &(&), 



= sech c= sech a sech b. 

 A + B<tt/2. 



Many readers will perceive at a glance that these are 



* Bull. Geol. Soc. America, vol. iv. 1893, p. 13 ; and Amer. Journ. 

 Sci. vol. xxiv* 1907, p. 1. 



