traced upon the Surface of the Sphere. 



341 



taking. I should think it a task of some difficulty, in the then state of 

 mathematical science. However this may be, the point is fully established 

 in the present article ; and we see, that, in order to return to the same 

 point, the ship must pass through both poles, and make two successive sets, 

 each of an infinite number of convolutions round each of those poles, before 

 it returns. We need not pursue the subject further here. 



XXXIII. 



To find the projections of the loxodrome on a plane parallel to the 

 equator. 



By (XV. 2.) we have the value of v, the radius vector, in terms of <p, 

 and the constants of the projecting data ; and by (XXXII.) we have the 

 value of </> in terms of 6. Combining these two, we have 



' 



0-) 



Or, 



J 

 v 



2a(b + 



(2.) ' 



* This is the general equation of COTES'S spirals : hnt as it does not admit of the va- 

 lues k = " * ~ l , the sixth species cannot be formed from the rhumb line by pro- 

 jection on a plane at right angles to the axis of projection. All the others can. See also 

 HYMERS'S Geometry of Three Dimensions, p. 136. 



VOL. XII. PART I. XX 



