82 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



Hence C2 = o, and the ratio K^:K. is determined. From (7) and 

 (9) we have the approximate equations 



ct, \ fe'+l/ 



Hence K^ ^ +K.= —cu and 

 2 



cot Co ^.+ j Co ^^.^, 



As the solution for Z^ reduces to a constant by C.^o, the value of 

 </) may be taken as merely /xZ^ in (8) ; or 



.=/x I 



en 



COt-^Co- ^^ 



■ t tan-^ C - I + C (tan-i f^ + ^^-:^) 



We are dealing only with very small ellipsoids and hence Co is 

 small. Hence ^ reduces approximately to 



^= f^[-Ctan-^C-i + 2a]. 



TT 



2 



The velocity along the ellipse C=Co may be obtained like the normal 

 velocity. The element of arc ds is 



ds='\/ dx^ + dr^ — c 



I — jU. J \ I — jU, 



C?<^ _2U \ I-^^ PI 



rf.y - TT \Co^ + /.-^'- -•• 



The value of this velocity is greatest when ju,=o, i. e., at the ends of 



OH 



the ellipse, as was to be expected ; and its value is then "^ . The 



value of Co iiiay be expressed in terms of the axes of the ellipse. 

 From (3), 



le is Co = 

 approximately 



Indeed the value is Co — — = approximately. Hence the velocity is 

 c (1 



Maximum velocity = — -7-. 

 TT 



By Bernoulli's principle for a stream line or Kelvin's theorem on 

 irrotational motion we have 



" H z;-=:: const. 



P 2 



