traced upon the Surface of the Sphere. 317 



Hence (3.) becomes 



sin 2 i cos 2 i = cos 2 a sin 2 i cos 2 <f> + sin 2 a cos 2 i sin 2 6 sin 2 ^>, (8. ) 



From which 





sin i \/ cos 2 i cos 2 a 



.(8.) 



' sin 2 a cos 2 i sin 2 6 cos 2 a sin 2 i 



The + and as before referring to the two branches of the curve about 

 the two poles. 



FIG. 20. 



Recurring to the formulas for orthographic projection (XV. 5.) we 



have 



2 J ^ 9 -/. ^ ** 1-9/3 V 2 



sin 2 (f) = - ; cos 2 </> = ' and sm ^ = 4 



These inserted in (7.) give 

 ?/ 2 (sin 2 i cos 2 a cos 2 i sin 2 a) + a? 2 cos 2 a sin 2 i = a 2 sin 2 i (cos 2 a cos 8 ) (9.) 



Now, since i is always greater than a, the factors of the equation (9.) are 

 essentially +, for they may be written, 



7/ 2 sin (i a) sin (i + a) + n 2 cos 2 a sin 2 i = a 2 sin 2 i (sin i a) sin (i + a) (10.) 



This projection, therefore, is an ellipse, whose major and minor semi- 

 axes are respectively 



+ a sin i, and -H a\/ 1 cos 2 i sec 2 a. 



VOL. XII. PART I. 



s s 



