181 



r -J- 1 rdr, 



{' + ''%) " 



F (r, w) = being the polar equation of the given curve. 



It is convenient to distinguish the curves of the two series 

 by calling those of the former positive, and those of the lat- 

 ter negative ; vre may also generally denote their polar co- 

 ordinates by the symbols rj^^, w_i.^. 



If the given curve, which may be denominated the base 

 of either system, be an ellipse whose centre is the origin, it 

 will be found, by applying the above formula, that the nega- 

 tive cui'ves will in general have their arcs expressible by 

 elliptic integrals of the first and second kinds, whose modulus 

 is the eccentricity of the base-ellipse. The arc of the first 

 will involve only a function of the first kind : a result which 

 has been given by Mr. Talbot in a letter addressed to M. 

 Gergonne, and inserted in the Amiales des Mathematiques, 

 torn. xiv. p. 380. 



A function of the third kind, with a circular parameter 

 — 1-1-6*, where h is the semiaxis minor of the elhpse, its 

 semiaxis major being unity, and the modulus of which is the 

 eccentricity, enters into the arcs of all the positive curves; 

 and their general rectification depends only on that of the 

 ellipse, and of the first derived, both positive and negative. 



The quadrants of the eUipse, and of the first two curves, 

 positive and negative, are connected by the following rela- 

 tion : 



(s_i + si) s_i = (3 s — s_2) (2s — Sa). 



It is worthy of notice, that if the eccentricity be .^ ~]., 



the functions of the third kind disappear, and the rectifica- 

 tion of both series depends only on that of the ellipse and of 

 the first negative curve. 



