JoLY — The Associative Algebra applicable to Hyperspace. 79' 



Similarly, for odd spaces, 



D.tti = 2 (/5 cos cs + ^' sin cs) ; 

 and on integration, 



«! = ^i(«"i cos CiS + 4 sin Cis) + ^2(4 cos CjS + 4 sin Cos) + . . . 



+ 5„,(«2,„-l cos C,„.S + 4,„ sin C„,S) + 3,„+l?2m+!, 



■with the condition 



512 + ^3^+ . . . + 3,„+i2^ 1. 



As before, the condition Ta^ = 1 reduces the constants of integration, 

 to this form. 



Finally, as ai = D,p, the vector to any point on the curve in even 

 space is 



hi . . . . - 



P = Po + — (- h sm CyS -f ?2 cos CiS) + . . . 



K, ... 



+ — (- hm-i sm c„,s + «2„, cos c,„s), 

 ^»» 



and that to any point on the curve in odd space is 



ii . . . . , 



p = po + — (- ^1 sm <?iS + t2 cos CiS) + . . . 



+ — (- «2m-l sin (?,„S + 4„ cos C,„S) + i,n+lhm+lS' 



¥ox the curve in even space the distance of any point on it from the 

 extremity of po is constant, or 



This curve is, perhaps, more analogous to the circle than to the helix. 



8. Reverting to the formula of differentiation for moving axe& 

 which was given in the 6th Article in the form D^w = FiOw, I shall 

 inquire what the quantity O becomes when expressed in terms of the 

 vector units /i, «2, . . . «'„, introduced in the last Article. This is an 

 example of a transformation from one set of unit vectors (a) to another 

 set («"), both sets being mutually rectangular. 



A verification of the simplest kind (consisting merely in the appli- 

 cation of the formulae «'i4 . 4 = - Hj Q-nd «i4 'H = - hh • h = h) shows 

 that the equation D,ai = ViQai is true, provided ai is one of the vectors 

 thus denoted in the last article, and provided also the quantity O is 

 defined by the equation 



fl = Ci?i?2 + <^2*3^4 + . . . + C„j?2m-l*2in' 



This is true, whether the space is of even order {2m), or of odd order 

 (2m + 1). 



