336 Mr DAVIES on the Equations of Loci 



Hence we have log tan- = c; or since tan - = 1, log tan- = 0, and 



4 44 



equation (3.) becomes simply 



log tan = 6 . cot a ..... (4.) 



This equation may be rendered more convenient for subsequent investi- 

 gation in some other forms. 



Take log" 1 of both sides, then 



tan |0 = fcota.0 = K 6 



(where 6 is the hyp-base, and K e cot ). 



Or squaring both sides, and expressing tan 2 \ in terms of cos 0, it becomes 



1 cos 



&<> = = -- 



1 + cos 



Or reversing this result, we have 



,., 



cos = -- = -- ......... (5.) 



I+ K 2 K-6 + K<> 



Also, sin0 = -4- 2 *' _ -+- __ 2 _ (6.) 



-!+' -K-' + K* 



Q If 6 O 



tan0 = _^ _ = -f- - ! _ ... (7.) 

 1Kte - K - K t 



And so on with other functions of the spherical radius-vector 0. 



XXXII. 



We proceed to ascertain the course of the curve, which the equation en- 

 ables us easily to perform. 



The discussion divides itself naturally into two cases, according as cot a 

 is positive or negative, that is, as a itself is greater or less than a right 

 angle. We shall keep the two cases on the same folio, opposed to one an- 

 other, for the sake of more ready comparison. Moreover, as the variable 

 is expressed by means of its tangent, there are, for every value of 0, two 

 points in the locus, diametrically opposite to one another on the sphere, and 

 hence we shall first trace that which is given by the least value of 0. We 

 shall also take a only with first and second quadrants, as, by taking it in 

 the third and fourth, we only get a repetition of the same angle (measured 

 in the opposite direction, it is true, but upon the same great-circle tangent 

 to the curve), and hence only a repetition of the same results as are ob- 

 tained from the first and second. The value of 6 = 0, is placed in the 



