422 Br. Silvanus Thompson on 



Multiplying (1) by sin ^, we get 



^ sin (f> } = sin 9 . sin ^ x . ...... (4) 



Squaring (3) and (4), and adding them gives us 



(I) 2 + ( fj - 2 ft cos *' = &in2 h ° ' (5) 



This is the equation to an ellipse, such as is represented by 

 fig. 2 (PI. VI.). According to the values given to $! there 

 arise three principal cases. 



Caseii.). K 



, IT ,->7T 



** = 2 ° r 3 2 ? 

 then 



sin (pi = ± 1 and cos <£j = ? 



and the equation becomes 



X 2+ Y? 



This is the equation to an ellipse set orthogonally with 

 respect to the coordinate axes as in fig. 3 (PI. VI.). 



Case 



(ii.). If 

















&=o, 



sin $> x = 



o, 



and 



cos 



*i= 



= i, 



and the 



equation becomes 

















x 2 f 



— 



2 x Yl 



= 0, 







whence 





y^ 



+ 



I 1 "' 









which is the equation to a straight line into which the ellipse 

 shrinks as in fig, 4 (PL VI.). But its length is limited by 

 the prior expressions, since x and y cannot exceed X and Y! 

 respectively. 



Case (iii.)« If 



tf) 1 ~7r, ein^i^O, cos$i~~ *lj 



