An introduction to the mathematical theory of attraction ...

Paraboloids of Revolution. 189
we get a=O, = q,, 7 = b. Hence, wehave
x = -(k + h) + ~-(k - h) cosh a,
= k + h) - ( - h) cos,. (56)
v = (k + h) - (k - h) cosh Y
104. Corresponding Points.-Two points whose coordinates are x, y, z and x', y', z', situated on confocal paraboloids of the same family whose parameters are À and À',
correspond when
_\ - 'y y' z z '
x A-$ (x -,) - ('-h)' - /( -k) = (-K)'
(57)
If P2 and P2 be two points on. a paraboloid, and P'1 and
P'2 be the corresponding points, then P1P'2 = P2P'1.
For
iP'22 = (X - X2)2 + (y' - y ') + (z - ')2
= x' - x2 + 2(X - ÀX) + y h - y2
+ {I( k) \ J( z) - 2 2
= ('1 -,)2 + (y' - y2)2 + (z'_ - z2)
yX'-h X'-k X- h -
+ 4(À - ')(', -, + À - ') = P'iP.
By supposing À to vary in equations (47), whilst i and v
remain constant, it appears that the line of intersection of
the surfaces whose parameters are u and v meets confocal
paraboloids of the remaining family in corresponding points.
L Hence, when the equipotential surfaces of a field of force
are confocal paraboloids of the same family, corresponding
points lie on the same line of force.
105. Paraboloids of Revolution.-If h = k, the two
systems of elliptic paraboloids become systems of paraboloids