...attraction of the mass whose density at the point (x, y, z) is p ; or, in other words, (2) where r is the distance between the points (x, y, z) and (x, y, z), p the density at (x', y', *'), and the limits are — oo to +00 , is the complete integral of (1) subject...

...formula for the distance between any two points in space. 93. Let it be required to find a formula for the distance between the points (x' , y', z') and (x", y", z"). If three planes be passed through each point parallel to the co-ordinate planes, they will form by...

...z) be any variable point on the line. By Art. 169, x — x', y — y', z — z' are the projections of the distance between the points (x', y', z') and (x, y, z) on the axes ; and since this distance is equal to its projection on either of the axes divided by the...

...vortex of this strength, situated at the central line of the vortex core, are given by w = s where r is the distance between the points (x, y, z) and (x, y, z), and the integrals are taken all round the vortex ring. Now X = p COS i/r = a COS i/r + an COS rn/r...

...point (x, y, z) of the surface g^-1 9r— 1 dr~ l to — -^— , — ^- - , — ~r — , where r is the distance between the points (x, y, z) and (x', y', z') ; and we can find A so that the surfacetractions may be given functions if we can find displacements...

...space. The notation is as usual. It is only necessary to say that H is a function of the time only, r is the distance between the points x', y', z? and x, y, z ; c = du/dx + dv/dy + dw/dz ; d/dt is put for differentiation which follows a particle of the fluid,...

...y2 - fc2z2 = 0 (10) where the constant is introduced in a form to insure the reality of the conic. The distance between the points (x, y, z) and (x', y', z') is then, when written in full dist xx' = ., (xx' + yy' - k2zz') .,,. ik arc cos — . . /— • (11)...

...direction ratios a,b,c are abc Any point on it is taken as (*l + ai, ?! + ht, z\ + et). Note, 't' is the distance between the points (x¡, y,, z¡) and (x, y, z) if a, b, c, are dc's of the line. Vector equation of a lin« passing through the point A(o) and parallel...

...ratios a, 6, c are yy\ abc Any point on it is taken as = t. (xi + at, yi + bt, z± + ct). Note, 't' is the distance between the points (x¡, y,, z,) and (x, y, z) if a, b, c, are dc 's of the line. point r = a +Xb for scalar X. Vector equation of a line passing...

...+(y- j3)2 + (z- y)2 = a2, where a is positive or zero. And as the left-hand side of this equation is the square of the distance between the points (x, y, z) and (a, /J, y), this shows that the real locus is a sphere in the ordinary sense of that term, with centre...