If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

00on the right-hand side is such that the coefficient of x n ix n Zx n 3...

00If therefore we choose a quantity e such that log e I o X X= I, log i oe = X, which gives (by more accurate calculation) e=2.71828..., we shall have lim(loge(I+0))}/0=I, and conversely 'lim' {ex+0 - e x } 143= The deduction of the expansions log e (' +x) = x - Zx 2 + 3x 3 - ..., e x = I +.x+x2/2!+x3/3!-}-..., is then more simply obtained by the differential calculus than by ordinary algebraic methods.

00When the plane zx is not a plane of symmetry, we have to consider the terms in xy, 2 y, and y 3 .

00Kirchhoff'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.

00Since loge(I +x) =x-2x 2 -3x 3 - 4x4+&c., we have, by changing the sign of x, log e (I - x) _ - x - zx 2 - 3x 3 - x 4 - &c.; whence g 1 +x to=2(x+ix'+1x5+&c.), e l - x and, therefore, replacing x by p +q, log e q =2 p +q +3 () 3T ?

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').

00To find the pressure exerted by a bar AB on the pin A we compound with the force in AB given by the diagram a force equal to P. Conversely, to find the pressure of the pin A on the bar AB we must compound with the force given by the diagram a force equal and opposite to P. This question arises in practice in the theory of three-jointed structures; for the purpose in hand such a structure is sufficiently represented by two bars AB, BC. The right-hand figure represents a portion of the force-diagram; in particular ZX represents the pressure of AB on B

00The most simple example is in the two systems of equations x': y': z' = yz: zx: xy and x: z'x': x'y'; where yz =0, zx =0, xy = o are conics (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system.

00The code is currently at beta... asp - amiga SPectrum emulator aSp is a Sinclair ZX Spectrum Emulator for the amiga.

00Originally known as Ashby Computer Graphics and later Ultimate Play The Game, Rare immediately established a reputation for graphically impressive games, many for the Commodore 64, ZX Spectrum and the Nintendo Entertainment System.

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