98 



THE TIDAL PROBLEM, 



where c' depends upon the new law of density. We have in the units defined 





(51) 



m 



the Ci, — ^, m^, M, P^, and R^ being given in equations (22), (40), (43), 



TO 



(44), (46), and (47) respectively. 



With this value of c' we must determine a new value of /j., say //', from 

 (21). To faciUtate the solution (21) may be written in the form 





(52) 



We may draw the graph of this function. It is the sum of two functions 



(53) 



Vi 



and 



2/2 



3 "" ft'' 



4 tan fi' 

 ^3 (tan/—/) 



(54) 



Fia. 12. 



For / = we see that t/i= — ooand 2/2= +<^> but that 2/1+2/2 = 2 — c' = 1.98. 



The curve for y^ is very simple. The value of y^ is positive while [x' varies 



from to 7c, vanishing at //' = ?:. Then 1/2 becomes negative and remains 



negative until 2/2= — "^ by the vanishing of tan / — //' at / = 257° 30'. 



4 Ztz 



Then 2/2 changes sign and descends from + 00 to - at /=— and to zero 



