442 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If we consider a field of 1-vectors, that is, a 1-vector function f 

 of position in space, we are naturally led to enquire what meaning, 

 if any, should be associated with the formal combinations 



V'f and \/xf 



obtained by operating with the 1-vector V. Let 



f (.ri, X2, Xi) = /,ki + f-iko + /4k4. 



Then 



OXi d.1'2 0X4 



^x' = (f - fr) ^^^ + if + f >" + if + f) '^'-'- 

 \dxi dxoj \dxi dXiJ \dxo dxj 



Of these the first, V • f , is a scalar function of position, and the second, 

 V>f, is a 2-vector function of position. They correspond respectively 

 to the divergence and curl in Euclidean three dimensional space. 

 The first, V -f, has indeed the same form as usual. And this was to be 

 expected: for physically' or geometrically the idea of divergence 

 depends on translation alone and not on rotation, and it would also 

 have appeared analytically evident if we had used in the definition 

 of divergence the operator V* instead of V. The second, Vxf, 

 differs from the ordinary curl not onh^ in that we have retained it as 

 a 2-vector (instead of replacing it by the 1-vector, its complement, 

 as is usually done in Euclidean geometry of three dimensions), but 

 also in that it represents non-Euclidean rotation in the vector field in 

 the same sense that the curl represents ordinary rotation. 



If F is a scalar function of position, then V F is a 1-vector function. 

 We may then form 



Of these the second, VxV F, vanishes identically, as may be seen by 

 its expansions or by regarding it as an outer product in which one 

 vector is repeated. The first, V • V F, may be expanded as 



o.tr 0x2' oXi- 



and V* V corresponds to Laplace's operator in Euclidean geometry. 



If f is a 1-vector function, there are four different expressions which 

 involve the operator V twice, namely 



VV-f, V-Vf, V-Vxf, VxVxf. 



