Without attempting to answer this question categorically, it may be pointed out that within the limits of the family (Ptychoderidae) which is especially characterized by their presence there are some species in Y art dY **YY** cts, posterior limit of collar.

In general we may have any two forms 01/1X1+ 'II ï¿½ **Yy** + 02x2) p Y'x =, / / being the umbrae, as usual, and for the kth transvectant we have (4)1,,, 4)Q) k = (4)) k 4)2 -krk, a simultaneous covariant of the two forms. We may suppose of, 4, 2 to be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants.

In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 **yy** y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?

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.

The equations to the tangent and normal at the point x' y are **yy**' = 2a(x+x) and aa(y - y')+y'(x - x')=o, and may be obtained by general methods (see Geometry, Analytical, and Infinitesimal Calculus).

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').

And 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').

Bay ll c t ' LY Q " **yy** be Pony.