462 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



planoid. The line 00' = k represents the lapse of time between 

 and 0' ; and this is equal in magnitude to O'Q or Ig, the space compo- 

 nent of 1. The interval OG = k^l — v^ and the interval O'G = kv = l^v. 

 The direction w projects into the direction v. Hence as a vector, 

 O'G is equal to l^w. The quantity ls*v = O'F may be obtained by 



r< 



o 



'l'^ 



IV 



dropping a perpendicular from G to O'Q. The interval FQ is then 

 Is — ls*V or ^4 — Ij'V, the expression which occurs in the denomina- 

 tors. The vector GQ = r is clearly 1^ — /sV or 1^ — /4V. 



44. We shall now remove the restriction that the (5)-curve which 

 gives rise to the potential p = -w/R = — w/Cl-w) is rectilinear, and 

 consider the general case of any (5)-curve. For the sake of simplic- 

 ity in this complex problem we shall use dyadic notation (see appen- 

 dix § 61, ff.). The results, however, might all be obtained by mean& 

 of the more elementary geometric and vector methods. 



We may write 



Op = 



w 

 R 



^^V+i^o-^-i^o^^^ + ^O"- 



Now 0>w is defined so as to satisfy the relation rfr«Ow = dyf. A 

 displacement (Figure 23) dr = "W ds parallel to w, makes a change 

 f/w = cds. A displacement dr along the vector 1 (Figure 24) intro- 



