WILSOX AND LEWIS. — RKLATIVITV. 441 



If we select three mutually perpendicular axes ki, k^, ki, and denote 

 by .ri, X2, Xi the coortlinates (intervals) along these axes, then 



dF dF OF 



dF = dxi . + (/.r2 „ + (Ixi ^ = {dxiki + dx->k2 + dxik.i)'VF. 

 oxi dx> oxi 



From this V may be determined as 



V = k,/- + k,/ -k,/ • (40) 



dxi dx-i dXi 



Thus V appears formally as a 1-vector, and may be treated formally 

 as such.^° 



direction of dr and where ds is the interval or magnitude of dr, we may write 



dF 



dF = dsVi'\^F or u«vF = --• 



ds 



Hence the component oi V F along the direction dr is the directional derivative 

 of F in that direction. Consider now two neighboring surfaces of constant F. 

 Suppose first that the (approximately parallel) tangent ])lanes to the surfaces 

 are of class (7), so that the perpendicular V^ is a (sj-vector. Then, in 

 our geometry, the perpendicular from a point on one surface to a point of the 

 other is, of all lines of its class, the line of greatest interval ds (§12). The 

 directional derivative along the normal is therefore numerically a minimum 

 (instead of a maxinuun) relative to neighboring directions. In fact, the 

 derivative along a line of fixed direction would be infinite, because along the 

 fixed cone ds = 0. Along the (7)-hnes the directional derivative varies 

 between and 00. Suppose next that the tangent planes are of class («), so 

 that the perpendicular vi^ is a (7)-line. Then the interval ds along the 

 perpendicular from a point on one surface to a point on the other is neither a 

 maximum nor a minimum, but a minimax. For it is less than along any 

 neighboring direction (of the same class) which with the perpendicular 

 determines a (7)-plane, but greater than along any neighboring direction 

 (of the same class) which with the perpendicular determines a (5)-plane. 



30 The above definition of V F depends on inner nudtiplication, and hence 

 upon the notion of perpendicularity or rotation. It is, however, interesting 

 to observe that we may define a differential operator v' dependent upon the 

 outer product, and hence upon the idea of translation alone. The definition 

 would then read 



a^hxcdF = drxvF = (,a,dxi + hdxi + (idx3)y.V'F, 



where a, b, C are any three independent vectors, and where Xj, Xt, Xt are co- 

 ordinates referred to a set of axes along a, b, C. Then 



v = bxc V, — h cxa^,^ — I- axb ^r-- (41) 



axi dxi 6x3 



Now v' may be regarded as a 2-vector operator in tlie same sense as V is 



regarded as a 1-vector. To show the relation of v' to v. when the ideas 



of perpendicularity are assumed, we may take a, b, C as ki, ks, k4 and Xa as 



Xi. Then 



v = k., .— 4- k4, -- +knjr = kiv- + k2 t; k, ^— I • 



oxi dxi dXi \ dxi dx, axtj 



Thus v' is the complement V* of V. In fact if 



(dF)* =drxv'F and dF = dr-vF, 



our rule of operation (30) shows that v' = V*. 



