traced upon the Surface of the Sphere. 263 



Society of Edinburgh has done me the honour to insert in its 

 Transactions ; and at a future time I hope to confer the same 

 simplicity and completeness upon some other questions which 

 have been repeated subjects of unsatisfactory discussion by every 

 other process. My present purpose, however, is to lay down a 

 sketch of the systematic principles of the method, to furnish so- 

 lutions to the problems that naturally present themselves at the 

 outset of the inquiry, and to apply it to the discussion of several 

 problems, that, by their repeated agitation, have acquired a cele- 

 brity which gives them a high mathematical interest, independ- 

 ent of any practical utility they might be supposed to possess in 

 the affairs of common life. 



The method which I propose to employ is an equation between 

 two spherical variables. Sometimes the latitude and longitude, 

 sometimes the polar distance and the polar angle, will be the more 

 convenient. These two cases have a much closer analogy on the 

 sphere than the methods by polar and rectangular co-ordinates 

 have in piano. The former is that generally employed in the 

 hour-lines ; the latter will be employed almost exclusively here, 

 the class of problems here discussed seeming generally to admit 

 of more ready treatment under this than the other form. The 

 methods are, however, readily convertible, so that we can pass and 

 repass from one to the other without that difficulty and com- 

 plexity which commonly attends such transitions in the case of 

 rectilinear co-ordinates. All this is very simple, and will probably 

 have often occurred to mathematicians before : yet there is still 

 only one single exception to the statement, that no direct and 

 promising attempt has ever been made to put it in practice *, 

 and even that was dropped in the very outset. The great diffi- 

 culty amongst the earlier geometers was found to be the incon- 

 venient notation by which trigonometrical relations were im- 



* See Note E. 



