648 Mr. F. Tavani on 



complex quantities are complex of second order or vectors — 

 that is to say, the equations o£ group (i.) say that each 

 complex is perpendicular to all those which precede it in the 

 given sequence ; while the first equation of group (ii.) 

 expresses the projection of b' upon a, the second equations 

 express the projections of c/ upon a and b, the third 

 equations give the projections of d' upon a, 6, c, and so on. 

 The quantities a, 5, c, d ... may be assumed further to 



verify the conditions: a 2 = l, 6 2 =1, c 2 = l 



and 



mod. a" mod. c 





_a[_ V c= d' 



mod. a! ' mod. V ' mod. d' ' }■ . (iii.)' 



mod. a" mod. b n mod. c" ~~ mod. 



which are obviously equivalent to those of the following- 

 group : 



a' — bmod.a, c' = bmod.c / , ~\ 



a' = cmod.a\ b' = cmod.b f , d' = cmod. d',r - n -i) 



Let P be a point moving in a space of n dimensions as a 

 function of a variable t (time), so that 



mod.P' = r (speed of P) 



and 



P' 



a= — , an equation analogous to those of the group (iii.). 



Let also ir 1 tt 2 tt z . . . ir n indicate points, such that the 

 complex (ir l — o) has the direction of P' and passes through 

 the fixed point o. In the case of complex of second order, 

 and P moving in ordinary space, 7r 1 describes, while P 

 moves, what is called the spherical indicatrix of the tangents 

 to the path ol P. We assume the point o to be such that 

 7r 1 = o + a, and in a similar way 



7r 2 = o + b ) 7r 3 = o + c, 7r 4 = o + «f. . . (iv.) 



In the case of complex of second order, vectors, and P 



* The meaning of these equations is obvious : they express that the 

 complexes are of modulus =1, and each perpendicular to its derivate. 



