X = c cosh ^ cos i^j, y = c sinh <^ sin ^, [61b, c] 



dw Uz \ "^ 1 1 



dz \dwi c sinh w cfsinh fp cos iIj + i cosh <^ sin ^i) 



( (fit I ^ 1 



= , G = sinh^ <;S cos^ ^ + cosh^ <^ sin^ ^, [61d,e] 





2 



G = sinh2 <^ + sin2 ^/, = i^(cosh 2<?i- cos 2^/), [61f] 



by the use of hyperbolic formulas listed in Section 32. 



By substitution it can readily be verified that the solutions of [61b, c] for 9!) and ip can 

 be written 



cosh = — i\_{x + C)2 + y2] '/= + [(^ _ c)2 + y2] ^'\ [gig] 



COS rf, = —/[(a; + C)2 + y2] ''^ _ ^(^ _ ^^ + y2] ^| [glh] 



Here the positive square root is meant. The sign of 96 and the value of ijj must be chosen to 

 fit [61b, c]. 



Singular points occur wherever both <p = and sin i/f = 0, so that dw/dz ■* 00, hence at 

 (c, 0) and (-C, o). Furthermore, two types of multiplicity occur: ip is many-valued with a 

 period of 2n-; and the same point on the s-plane corresponds to -<^j, -i//j, as to ^Sj, i/(^. 

 The latter multiplicity extends to dw/dz, which has opposite signs for -^S^, -i/^j, and for c^^, 



By elimination of <poT i// it is found that 



■^ a;2 w2 



= 1, ^ =1. [61i,j] 



c2 COsh2 c2 sinh2 c2 COs2 i/, c2 Sin2 i/r 



Thus the curves <p= constant are ellipses, while the curves tp = constant are hyperbolas; 

 both families of curves are confocal, with common foci at (ic, 0), and, as usual, they cut each 

 other orthogonally. They are illustrated in Figure 93, also, in more detail, in Figure 129, on 

 which ^ may be identified with rp and 7; with i/t. Two of the hyperbolas degenerate into parts of 

 the a;-axis: 



ip = 0, X = c cosh <P> c; tp = n, x = - c cosh^ p< - c, 



using [61b, c]. Again, on the y-axis, x/j = n/2 or 3n-/2 and y =- c sinh 96. The ellipse for ^ = 

 degenerates into the a;-axis between ± c, on which x = c cos ip. 



137 



