traced upon the Surface of the Sphere. 419 



We shall next proceed to investigate its course, and we shall find that 

 (5' 6') represent two branches, not continuous, but symmetrical with respect 



to the equator. 



t 



2. Take (5') ; viz. tan = 6 cot ' ' . Then, 



(a,). Let = 0; then tan = 0, log tan = : and as e -^ 1, all its 



powers are greater than 1 ; and hence 6 = ^ , or the curve winds round 



the pole negatively to infinity or the pole is an asymptote. 



(&,). Let $ = ; then tan , and log tan = - , or the equator 

 * . 



is an asymptote to the positive branch of the curve. 



(c,) At all intermediate positions, the value of 6 is finite, and either po- 

 sitive or negative. 



The branch of the curve, then, designated by (5') winds round the pole as 

 a spiral in the negative direction from the origin of 6, and continually ap- 

 proximates towards the equator on the positive side of the same origin ; but 

 never reaches either the pole or the equator during any assignable number 

 of revolutions. 



7T 



(d t ). If <p -^ ; then tan = p, and log tan is imaginary ; that is 



the branch denoted by (5') does not reappear beyond the equator, or does 

 not cut it after infinitely revolving round the sphere. 



(e t ). There is, however, a secondary branch involved in the same equation, 

 for tan = tan (TT + 0) ; and hence (5') is fulfilled by this latter system of 

 values of as well as by the former. At this point, however, the subtan- 

 gent a is changed from a to a + TT, and the second curve is that which is 

 traced out by the opposite extremity s' of the diameter passing through the 

 current point s of the first curve. The second curve, or the conical branch, 

 as I have elsewhere called it, is not a solution of the same geometrical 

 problem, though it is of the same algebraic equation. They are not, strict- 

 ly speaking, branches of the same curve ; but they are solutions of the same 

 problem, if, instead of the datum being a given arc, it had been an arc 



VOL. XII. PART II. 3 H 



