p = r — . 



and the Rectification of Curves. 213 



might be discovered by the investigation of the analogue of the 

 known formula for plane curves 



dr 



dp' 



Again, the sum of the curvatures at any point of a surface is 

 expressed by the formula, in rectangular coordinates, 



1 JL_ = _ (l + g a) r -2 p gg + (l+jP 8 )f . 

 Ri R 2 ~(l+p* + q*)% 



we have no corresponding expression in polar coordinates. Other 

 desiderata will readily suggest themselves. 



8. With regard to the rectification of curves, it may be useful 

 to make a few observations upon a subject which has recently 

 attracted much attention among French mathematicians. In 

 the Notes by M. Liouville to his valuable edition of the Applica- 

 tion de V Analyse a la Geometrie of the illustrious Monge, will 

 be found (p. 558) the following remarks : — 



" M. Serret a fait usage de certaines variables qu'il avait deja 

 employees au tome xiii. du Journal de Mathematiques, pour 

 resoudre le probleme suivant : x, y, z, s etant quatre fonctions 

 d'une variable independente 6 assujetties a verifier Pequation 



dx°- + dif + dz 1 = ds' 1 , 



exprimez sans forme finie et sans aucun signe u'integration, les 

 valeurs generales de ces fonctions. La solution de ce probleme 

 conduit, par exemple, a trouver dcs courbes a double courbure 

 qui soient a la fois algebriques et rectifiables algebriquement, ou 

 dont Pare depende d'une transcendante donnee. Le probleme 

 analogue pour les courbes planes depend de Fequation plus simple 



dx' 2 +dy 2 =ds 2 } 

 et se resout, comme on sait, par les formules 



y = f \0) cos d-f" \6) sin 9, 



s = ^'{$)+^"(d), 



ou la fonction i|r est arbitraire. Les formules de M. Serret pour 

 l'equation 



dz* + dy* + dz' 2 = ds' 2 , 



Bont beaucoup plus compliquees, et, pourtant, beaucoup moins 

 utiles." 



It appears to me that the integration of these equations may 

 be effected directly, and with great simplicity, by employing the 

 Calculus of Quaternions. 



Thus, in the notation of this Calculus, the first equation 



dxt + dy^ds* 



