

[ 647 ] 



LVT. Motion and Hyperdimensions, 

 By F. Tayani *. 



THE object of this note is (i.) to establish a general 

 relation showing how hyperdimensions are brought 

 into evidence by carrying out -an analytical operation which 

 is susceptible of dynamical interpretation ; (ii.) to derive 

 from the said relation, for the particular case of three 

 dimensions, an important characteristic which can be extended 

 to spaces of any dimension. 



Let us consider the two complex quantities 



a = (cii a 2 ... a n ) and b = {b 1 b 2 ... b n ). 



After having extended to them the method of representation 

 through the coordinates, we can assume as definition of the 

 scalar or internal product between them 



» 

 a/b= X (a A), 



n-l 



in which the first member represents the scalar product; 

 .according to Grassman's notation, the equation 



2(aA)=0 



expresses that the complex quantities a, b are perpendicular 

 ^to one another. 



Let us consider a sequence, a, b, c, d...n of n complex 

 ■quantities, of n coordinates satisfying the following 

 equations : 



bja — 0, ~] 



c/a = 0, cjb^Q, L . . . . (i.) 



d/a=0, d/b = 0, d/c = 0, | 

 ......... J 



(from which we obtain by derivation 



b'\a=-b\a', ^ 

 c'/a=-c/a' s <//b=-c/b' 9 [ < (ii>) 

 d'/a = - dia', d'/b =■■ - d/b\ d'/e = - d/c\ \ 

 J 



T the derivation being with respect to a variable t, of which 



the said quantities are assumed to be functions. 



The geometrical meaning of the equations (i.) and (ii.) is 



an extension of that which tbey obviously have when the 



* Communicated bv the Author. 

 2 U 2 



