136 Dr. L. Silberstein : Further Contributions 



The integration just applied to the standard conic can 

 obviously be extended to the " rectification l> of any curve 

 determined by <£(#, y) = or y=zf(w), the line-element, or 

 tensor ds of an infinitesimal vector being given by 



ds 2 = dx 2 -f dy 2 + 2 sib dxdy 



in case o£ any a, b as coordinate axes, and by 



ds 2 =dx 2 + df ...... (14) 



in case of: orthogonal axes. This subject does not call for 

 any further explanations. Nor does the proof that the arc 

 of any conic a:,., i. e. r 2 = ? ,2 = const, (homologous with k) 

 subtending the angle 6 is equal to 



Or, 



offer any difficulties. Thus also the whole circumference of 

 a conic K r (or of its reproduction obtained anywhere by the 

 transformation r' = r + const.) will be equal to 



2irr staudtians, 



no matter whether k v is still an ellipse or has already 

 become, say, a parabola (whose length would in the ordinary 

 treatment be infinite) . Hyperbolae may offer some peculiar- 

 ities whose discussion will be left to the reader. 



Polar coordinates r, 6 can at once be introduced, with 



,c = r cos #, y = r sin 6 



as their relations to the previous orthogonal x,y. The use 

 of such coordinates amounts to writing, for the position 

 vector of any point of the plane, 



r=r(icos0+jsin0), .... (15) 

 where i, j is a pair of orthogonal unit vectors. 



6. Extension to Three-Space. — In order to extend the 

 preceding investigation to three-dimensional space let us 

 take, instead of k, a non-ruled closed quadric (ellipsoid), 

 surrounding the origin O, as the standard or unit surface. 

 We will denote it shortly by Q. The T-plane will be the 

 polar plane of O with respect to Q. 



The equation of Q with any non-coplanar triad a, b, c as 

 -axes will be 



x 2 + y 2 + z 2 -f 26*23.?/: H- 2a n zx + 2a n xy = 1. 



