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Kirchhoff's expressions for X, Y, Z, the coordinates of the centre of the body, FX=y 1 cos xY--y 2 cos yY-{-y 3 cos zY, (18) FY = -y l cos xX -Hy2 cos yX+y 3 cos zX, (Ig) G=y 1 cos xZ+y 2 cos yZ+y 3 cos zZ, (20) (21) F(X+Yi) = Fy3-Gx3+i /) X 3epi.

00But the numerical factor appears to be yz'+zy', while it is the quantity yz' - zy' which really vanishes.

00For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factorlines are coplanar provided ij = - ji, for it would then assume the form ij(yz' - zy').

00He had now the following expression for the product of any two directed lines: xx' - yy - zz' +i(yx'+ xy')+ j(xz' '+zx') +ij(yz' - zy').

00And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion - + yy ' +2z') +i (yz ' - zy') +j (zx' - xz') +k (x y ' - yx').

00Also, if 0 be the angle between them, and x", y", z" the direction-cosines of a line perpendicular to each of them, we have xx' +yy'+zz' =cos 0, yz' - zy" = x" sin 0, &c., so that the product of two unit lines is now expressed as - cos0+ (ix" +jy" +kz") sin 0.

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