458 PROCEEDINGS OF THE AMERICAN ACADEMY. 



for Euclidean and for our non-Euclidean geometry. The equation 

 may be put in another form by the aid of rules previously given.^^ 



f d<T,*{)= f rf(T(,,i)*xO(). (65) 



In four dimensions a large nvmiber of special formulas may be 

 obtained by applying our operational equation to scalars and to 

 vectors of any denomination with either symbol of multiplication. 

 As examples we may write the formulas corresponding to Stokes's 

 and Gauss's theorems. Let p = 1 and apply the operator by inner 

 multiplication to a 1-vector function. Then 



JdS't = - JJiclS'0)'i = f j f/S.(Oxf). 



This is the extended Stokes's theorem. Again let ^ = 3 and apply 

 the operator by outer multiplication to a 1-vector function. Then 



///rfSxf = -////Ws-C»^f = -fffPziO-t). 



This is the extended Gauss's theorem, where fC represents a differ- 

 ential (pseudo-scalar) element of four dimensional volume. 



In these cases also the same equations apply in Euclidean and in 

 our non-Euclidean space. If, however, we write these two equations 

 in non-vectorial form, they become in the non-Euclidean case 



/ 



(fidxi + fodxi + fsdxs — fidxi) 



= //[ 



+ 



37 This equation embraces both of the operational equations given by Gibbs 

 in §§ 164-5 of his pamphlet Vector Analysis (1884) reprinted in his Scientific 

 Papers, 2. In case p + 1 is equal to n, the number of dimensions of space, 

 then (/^(p+i)* is a scalar and the equation has no meaning unless we adopt the 

 convention mxa = nia, where m is a scalar and a any vector. This convention 

 would lead to no contradiction, and might occasionally be useful. 



