180 G. PLARR ON THE ESTABLISHMENT OF THE 
This gives 
. “N : 4N ZN 
Up =7 cos xp + j cos yp + k cos 2p, 
; . IS TS 
where, for shortness’ sake, we designate by xp, yp, zp, under the sign cos, the 
angles which p forms with the three axes. We have also— 
A A A 
1 = cos *xp + cos *yp + cos *zp. 
This expression of Up gives U(— p) = — Up. 
We shall in the sequel, and concurrently with the system of the three rect- 
_ angular directions 7, 7, 4, make use of a triple rectangular system, whose vectors 
we designate by 
Up, Uc, Ur. 
Their directions are to be determined—1st, Up, by the given direction of p ; 
and 2d, Uo, by being drawn through O as a common origin of p, and of another 
given vector w, and in the plane determined by p and a, and so as to form an 
0 
Up Up 
Uc Us: 
angle with ow, of a positive value, and not exceeding aright angle; 3rdly, Uz 
will be drawn in respect to Up and Us, in the same relation of respective 
direction, as the direction of / is drawn in respect to the directions of 7, /. 
YN 4N IN 
In this case we shall have cos ow = sin pw, COs ta = cos 90° = 0; and the 
expressions for w to be considered in the sequel, will be of the two forms— 
UN Sein ZS 
w = Ta(Up cos pa + Uso sin pa), 
ao=ix+yjyt+ kz, 
Ta = Je +P t2. 
§ 3. Multiplication of Vectors one by another. 
According to the principle of the influence of the order of the factors, so 
happily introduced by the inventor of quaternions, we represent by pa, the 
product of the multiplicand a, by the multiplier p. 
In this manner the two products 
po, and wp, 

