470 Motion of a Particle on Surface of Smooth Rotating Globe. 



same longitudes as the most northerly points, are found 

 by solving 



zn 



or zn^ = # 0215i/r. 



The positive root is i/r = l-69. 



The distance from the equator of the double point is 



EX o dn(l-69) = 1330km. 



To find the width of the loop we proceed as in Case II. 

 above. From equation (9'27) the most easterly point of a 

 loop is given by 



dn 2 i|r-| = 0. 



This is equivalent to , 1 



and hence from the tables , 4*01 T ^ _.. 



Y= — q- K = *826. 



The difference of longitude between the most northerly 

 and the most easterly point of a loop, found by substituting 

 the value of ty in the formula (9*3) 



B</> = R\ (zn<^--0215f>), 

 is 241 km. 



It has been shown already that the displacement from loop 

 to loop is 150 km. Hence the width of a loop measured from 

 a point where the particle is going north to the next point 

 where it is going south is 482 km. ; whilst the distance from 

 the latter point to the next one where it is going north is 

 632 km. 



11. The formulse which have been used in the foregoing 

 investigation are only approximate, and it would not be right 

 to use them in higher latitudes. It is evident, however, that 

 the loops will become more nearly circular as the poles 

 are approached. The classification may be extended to 

 distinguish : 



Case III. a. — Loops not passing through or including the 

 pole. 

 b. — Loops passing through the pole. 

 c. — Loops surrounding the pole, pole not central. 

 d. — A circle of latitude. 



