454 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Thus we find, for example, that <0»<f is not always a plane passing 

 through f , and in fact will usually be a biplanar vector. Also <y" ^ is 

 not necessarily in the plane of F. 



We have used the same symbol <^ for our differential operator as 

 was used by Lewas in his discussion of the vector analysis of four 

 dimensional Euclidean space, and which corresponded to the "lor" 

 of Minkowski. There seems no danger of confusion, since it will 

 never be desirable to work simultaneously in Euclidean and non- 

 Euclidean geometry. Sommerfeld^* has also developed a vector 

 analysis of essentially Euclidean four dimensional space, and his 

 notation is an extension of that current in Germany for the three 

 dimensional case. For the sake of reference we will compare the two 

 notations, as far as the differential operator is concerned, in the follow- 

 ing table. 



OF ~ Grad F, 



O'f ~ Divf, 



Oxf ~ Rot f , 



0-F~ ®it) F. 



Operations involving <^ twice are of frequent use in a number of 

 important equations. These may be obtained by rules already given 

 if <0> be regarded as a 1 -vector. 



0«>F) = 0, (50) Ox(C»<f) = 0, (51) 



C>-(O-F) = 0, (52) Ox(OxF) = 0, (53) 



O-(O-3f) = 0, (54) 



0-«>xf) = O(<0-f) - (O-O) t (55) 



0-«>xF) = Ox(C>-F) + (O-O) F,35 (56) 



o-coxjf) = Ox(o-ff) - «>-o)r (57) 



The important operator 0*0 o^' O" ^^^^ sometimes been called the 

 D'Alembertian. In the expanded form it is 



c/.i'i" a.v-2- o.Vs~ oxr o.Vi- 



where V now denotes the Euclidean differentiating operator in the 

 ki23 space. 



34 Sommpifeld, Ann. d. Physik [4] 33. 649. 



35 Kraft (Bull. Acad. Cracovie A, 1911, p. 538) devotes a paper to the 

 proof and application of this formula. 



