38 G. PLARR’S ADDITIONS TO THE PAPER ON THE ESTABLISHMENT 
applying the distributive rule we get : 
p'La’ x (a'p’) ]=cos *u—h sin *w+7'(h+1) cos w sin w. 
The square of the tensor of this product will be: 
(cos *w—h sin *#)?+(h+1)’ cos *w sin *w. 
This expression being developed we see that the first power of f disappears, 
and the result will be, by h°>= +1: 
Tp'=’x(a'p’) P= (cos uw + sin 2u)?= +41. 
On the other hand, the product 
(p’a") x (@’p’) =(h cos w+7’ sin u)(h cos w—7’ sin wu) 
being effected by the distributive rule, and having: 
hP= +1, 7 
becomes, as we have seen (page 190) : 
= cos *w—f sin *w. 
This being a scalar, the square of its tensor will be 
cos ‘w+ sin *“w—2h sin “w cos *w 
= (cos *w+ sin *w)’—2(h +1) sin *w cos *w) ; 
so that : 
[T(p’a’) x (w’p’) P=1—2 (h+1) sin “a cos “uw. 
It appears therefore that the square of the tensor of p x|[a x (wp) | becomes 
equal to Tp*Ts*, whatever value we admit for f, provided h*= +1; whereas 
the square of the tensor of (pa) x (ap), im order to become = Tp’ x Ta* 
demands the value }=—1, to the exclusion of the value +1. 
Page 191, line twelve from below, correct vector into versor. 
Page 196, to the eleventh line from below, add: 
Provided that the versor of the product (¢+a) (+8) be the same as that of 
the product \ x [x (b+ 8) |, when a@+a has been assimilated to Ay. 
Page 197, after the italics in the middle of page, add: 
the verification as to the versor will result from the remark, at p. 200, on the 
- equality (a8) x (y8)=a x [B x (y8)]. 
The distributive rule of multiplication of two quaternions by one another is 
thus legitimated in so far as the tensor of the product is concerned. 
