154 THE CYCLIDE. 



If now 8 be a function of a, and <f> of ft, such that 



then 6 and < will be conjugate isothermal functions. 



If 6 = k I- -and< = /tf - (22), 



Jr-fecoso, Jr-ccoshft 



the condition will be satisfied. If r is greater than 6, we find 



tan . = (23). 



(r b')* b rcosa 



It is more useful to have a expressed in terms of 6, thus : 



J -&*)sin 

 sin a 



rcos 





- 

 C0 sa= - V . .............................. (24). 



If r is less than 6, we have only to write the hyperbolic functions of 



a a 



,. , ,,, instead of the circular functions of / t _j-\\ and to write (6 s j- 3 )* for 



(,--&')'. 



Similarly, we obtain for the relation between ft and <, when r is greater 

 than c, 



c + r cos h -r~- 



r + c cos h 



(25). 



j-r $-, 

 (r 1 c')* 



When r is less than c we must substitute (c 1 r 4 ) 1 for (j- 1 c 1 )* and turn 

 the hyperbolic functions into circular functions. 



