traced upon the Surface of the Sphere. 409 



that is, 



a _ cos 6 6, dr, + sin 6 6, rdQ, , ,_ , 



r~ / 'r*de + dr 



or, giving to a its value, viz. 



a r, sin tan~ * j^ 

 r.dV, 



(30.) 



we shall have (29.) converted into 



i _ cos e e, sin e e, r.de, , 31 , 



r r, r, dr 1 ' 



Whether the formulae (26.) and (31.) may ultimately assist us to a 

 greater extent than the methods now in use, in investigating the general 

 affections of polar curves, is not a proper subject of discussion for this note ; 

 but it may at some future period be inquired into more carefully. My own 

 employment of these formulae in a considerably varied number of trials, made 

 upon curves taken perfectly at hazard, leads me to think that they will be 

 found of great value ; but I have not entered sufficiently into a systematic 

 examination of the system to be able to say that there might not be ob- 

 stacles in the way of its universal application, that have not presented them- 

 selves in any of the particular cases which I have happened to select for ex- 

 periment. I think it, however, highly probable, that (as I have found to 

 exist on the sphere) there will be some cases where one method, and others 

 where the other method, will be most advantageous ; and as, on the sphere, 

 there does not appear to be any method of judging a priori which of the me- 

 thods will be most applicable, so also in piano, we shall be able to find no test 

 for deciding the question but actual experiment on the particular cases under 

 consideration. If nothing shall be gained in the way of general principles, 

 yet it will be of some advantage to be put in possession of an alternative 

 method of investigation, that offers considerable prospect of enabling us to 

 analyse the curve, when the common method shall, in that particular case, 

 either fail altogether, or else, by the complexity of the result, be thought less 

 elegant than the taste of the geometer may lead him to desire. 



