An introduction to the mathematical theory of attraction ...

Examples. 73
Again, if we describe a sphere round B as centre, we get from the spherical
triangle formed by the planes meeting at B,
sin OBA tan B= tan B, that is, tan 3=
r2 a
we have, therefore,
rz b b
w = tan-1 - - tan- -.
pa a
By Ex. 1, we have
v= gP + r2+ + alo r2 + r3 + a p + rg 4, b
p+r2-+c r r+r2 3a p+rr' b
Vc = log -- V = log - r2 + r3 - b = log- r3 - b'
also, from the geometry of the figure,
r22 = 2 + C2 r32 = r22 - a2 + b2;
hence
r2 + 0 (CV'2 + Vreî + V-r2 ) 2 + C
Ve = log = log
VI'2 - (Vr2 + c + Vr2 - c) Vr 2
A similar process being applied in the case of va and vb, we obtain
r2 + c r2 + a r3 + b
Vc = log, a = log -, vb =log
p r3 p
Substituting for v, Vb, V, and w, the values found above, we get finally,
(a r2 +4 c r + a
X=o-lo -log jr3
i r3 + b b 1r2+2 c)
y= log -+b-blog -j,
Z = tan-' -- - tan-' bp a a '
with the equations
C2 = ay2+ b2, 22 2 a2a2 + b2, 2 =p2 + b2.
9. Find the attraction of a homogeneous plane polygon at any point 0.
From O let fall a perpendicular O04 on the plane of the polygon; from A
let fall perpendiculars on each of the sides, and join A to each of the vertices.
We have then a number of right-angled triangles having a common vertex at A 4
find by Ex. 8 the components of the attraction of each of these at O. The
resultant of all these forces is the attraction required.
Most of the Examples in this Article are borrowed from Routh's "Analytical
Statics," vol. ii.