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.

But, assuming the distributive principle, the product of two lines appeared to give the expression xx' - yy' - **zz**' +i(yx' +xy')+j(xz' i j (yz' +zy').

He 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').

We may also write ur 1 = I +zu 1+ &c., since z is very small compared with u, and expressing u in terms of w by (25), (we find l 21- mv i fi(z) i I +z(c R w + ' R 2 w) do) = 27rmoti(z) I -f-**ZZ** (Ki + R2/ This then expresses the work done by the attractive forces when a particle m is brought from an infinite distance to the point P at a distance z from a stratum whose surface-density is a, and whose principal radii of curvature are R 1 and R2.

**Zz**, Septal strands.