cosh 3w-t - cosh w,t

8wz 32w- 32 w?(8) t

3coshd =3 -sinh w t t- -cosh 3w-t - 2 32w=8w! 32 w, 5 td = --sinh w t - 7cosh w t t

- sinh 3w t 32~2 8w- - 3 One can readily check that these solutions which are valid t o terms linear i n e, yield a Jacobian which is unity plus terms of order e2. Thus the Jacobian i s not exactly unity, although it i s correct through terms linear i n e. We now wish t o obtain a transformation from the center of a positive sector t o the center of the next positive sector. This involves folding together three transformations of the above form. For t h i s purpose the followtng formula is convenient. If one i t e r a t e s two transformations cf the form: x3(0) t a3x2(0)i(0)t a

1 = Cxco) + b

x3(0) t b3x2(o)k0) ( 9 ) x(2) = + + c2%3(l)tc x2(1)&(1) 3 + d2

3(1) ( and a

bl,el, d t are given by similar formulas where Re may thus obtain a transformation through one sector of the form of eq. 10, which although not exactly area preserving, is area preserving through terms linear in e. We now follow the procedure outlined in I, reducing this to normal form by means of the transformations: and and