Motion and Hyperdimensions. 649 



moving in ordinary space, 7r 3 describes, while P moves, the 

 second spherical indicatrix. 



From (iv.) we obtain by derivation 



■mod.Tr/ssmod.a', mod.7r 2 / = mod.^', mod.7r 3 / = mod.c'...(iv. 1 ) 



Let ns now introduce the notion o£ curvature pi through 

 the relation 



ds 



in which ds is the portion of the path described by P, and 

 da- x the portion of the path described by 7r 1? while P 

 describes ds. 

 Then we have 



, ( 



as \ 



ds\ 



dt) 



c/o-j /da l \ mod. 77V mod. a'* 



m 



Therefore mod.a'= — , and with a similar reasoning we 

 obtain Pi 



mod. b' 



P-2 



mod. c' = — , mod. d' = — 

 Pz Pi 



(vO 



where p 2 , p 3 , p 4 ... have a meaning similar to that of p x 

 established in a similar way by extending the same reasoning 



tO 7T 2 , 7T 3 , 7T 4 . . . . 



It is easy now to find the expressions of the projections 

 •of b' , c', ^'...upon the axis a, c, a, b } d, a, b, c, e, — 

 taken as systems of reference ; by replacing in the second 

 members of the equations (ii.), the values of a\ b' , c', d' ... 

 as given by (iii.j), and making use of the equations (v.), we 

 obtain : 



I b'la = — bla' = — bib mod . a' = 



I Pi 



I b'lc = - b\c' = - b/b mod. c' = - - 



v 

 Pi 



(2)^ 



c'/a: 



c'/b: 



■ c/a' = — c/c mod. a = 

 c/b' -— — c/c mod. b' = 



v 



P2 



c'ld= -eld' = - c/c mod. d'=-~ 

 111 p 4 



(vi.) 



