traced upon the Surface of the Sphere. 337 



middle of each column, and we proceed upwards and downwards from it to the 

 extremities of its possible values, viz. + inf. and inf., or + - and - *. 



The equation of the loxodrome is log tan J = cot a ; and as the re- 

 sult is given in tan <, the primary and secondary branches are diametrical- 

 ly opposite, or the genesis is of the conical kind. In what follows, let m 

 denote any real positive number, either whole or fractional, rational or irra- 

 tional ; and let K be any angle less than g. Then for the case of 

 a = o K > or a = -Q- K, we have cot a = + i. 



*** At 



FIG. 25. (See p. 80.) 



Continuing the same hypothesis, with the exception of what relates to a, 

 let us put a = !E -l- K. and a = 5 4. /. whence cota = -i. 



a = + K, anda = ?5 

 2 2 



FIG. 26. (See p. 81.) 



* The common form of the symbol of infinity is open to some objections. Baron 

 FOURIER, in his Treatise on Heat, and in his posthumous work on the solution of Equa- 

 tions, -has employed , and certainly it has the advantage of expressing distinctly the 



arithmetical origin and signification of that quantity. It has too,"some collateral advan- 

 tages, which, however, it would be foreign to our subject to discuss. 



