on an Oblate Spheroid. 337 



and multiplying this by 



the exterior 



71 cos 

 factor is 



a ' 



tan 2 /' 

 /tan 2 / 



cos 2 / 



COS 







Vi- 



-S 2 cos 

 tan 2 V 



/ 



and we have 



Vi- 



-S 2 ' 



A=:^j=p{«c«m,(n,c)-8*F JC }, 



which is the formula I used in the calculations. It would, how- 

 ever, have been better to reduce a step further; viz. we have 



sec 2 m i ( w ,c)= f ta 7 n/ ' ; {} 

 iV ' tan /cos/ l J 



= ^j— J! {J7r+F /C [F(&, 0)-E(fl, 0)]-E,cF(M)}+*>, 

 cos / 



and thence 



• sec 2 ra y (7i,c)-S 2 F /C 



Vl-3 2 



-^-{in + F^C a/1-8 2 cos/+F(M)-E(M)]-E,cF(M)}; 



CO& & 



or, finally, 



A=i7r + F,cF(6, 0)-F,cE(6,0)-F,cF(£, 6>) + VT^ 2 cos/F / c. 



It is easy with this expression of A to obtain the results already 

 found for the extreme values / ; = 0°, I' = 90°. 



As Legendre's Tables have for argument, not the modulus c, 

 but the angle of the modulus, say% (that is, sin %=c = 8 sin/), it 

 is convenient to replace \/l — 8 2 sin 2 / by its value cos % ; and the 

 formula? thus are 



cos^; 

 Asriw+F^Cv/T^P cos I+Y(b, 0) -E(6, 0)] -E,cF(Z>, 0), 

 where 



C=sin % =Ssin/, tan/' = */l-S 2 tan/, 0=90°-/'. 



And in the case intended to be numerically discussed, S=J ^3, 

 \/l — S 2 =i. I take /'as the argument, giving it the values 

 0°, 10°, . . . 90°, and perform the calculation as shown in the Table. 



