488 ON MULTIPLE ALGEBRA. [865 
61. The foregoing formulae agree with the theorem, No. 50, that the sum of the 
strokes 
\ = R O^Vi + + X 3 y 3 ), 
and 
= ^~7(T) ^ i7?i + fl - v ' 2 + ^ V 
is a stroke 
which is the diagonal of the parallelogram constructed with the given strokes A and 
/x; the proof is a little simplified by assuming (as is allowable) if = 1 and T = 1, in 
which case we have 
^ + /* = ^ K/^ 1 + a t l i) Vi + (pN 2 + oyx 2 ) r) 2 + (pA 3 + 07x3) 773}, 
from which the length and inclination may be calculated. 
As already appearing, the product x, y, z of any three points is the scalar 
rp rp np 
tvf ? 1^2 5 ^3 j 
2A> 2/3 
•^1) ^2 > ^8 
which is equal to the area of the triangle xyz, divided by that of the triangle 
A 1 A 2 A 3 . We have in like manner the product of a point x into a stroke A, viz. 
this is 
= (x^ + x 2 e 2 + x 3 e 3 ) (\ 1 rj 1 + + \ 3 Vs), 
which is = x 2 \ 2 + x 2 \ 2 + x 3 \ 3 , the two factors being in this case commutative; the value 
is equal to the area of the triangle formed by the point and the stroke, divided by the 
area A 1 A 2 A 3 . Of course, if in the one case the three points are in a line, or if in 
the other the point is in the line of the stroke, then the product is =0. 
62. We have yet to consider the product of two strokes; say these are 
^ = \Vi + A, 2 t7 2 + \ 3 7] 3 , and /x = /x^i + y 2 y 2 + y 3 y 3 , 
then we have 
A. y = (X1V1 + A 2 ?7 2 + \ 3 y 3 ) (fi x y x + n 2 rj. 2 + y 3 7] 3 ) 
= - {(X 2 yu, 3 - A 3 /x 2 ) 6! + (X 3 /Xj - Ai/xs) e 2 + (A^, - \ 2 y x ) e 3 }, 
which is a point regarded as having weight. If we take the two strokes to be 
A = (xy) = (x 2 y 3 - x 3 y 2 ) Vl + (x 3 y x - xpyA i h + {x x y, - x 2 y Y ) 7] 3 , 
/x = (xz') = (x 2 z 3 x 3 z^) Vi "b *i^s) V2 4" (®i®2 — x 2 z x ) V'ii 
and