Why Does a Ring of Dipoles Produce a Smaller Upward Force Than Expected?

[That Neuron]

http://books.google.com.au/books?id=x-XZBJngdM4C&printsec=frontcover#v=onepage&q&f=false

This post is referring to page 35-36.

I just find it odd that this book doesn't change it's mathematical description to align with the fact that a ring produces a component force up smaller than the force that it would produce if it weren't pointing straight up.

In other words, taking an infitesmal ring of small dipoles that interact with a single dipole of the same type above it by a force proportional to 1/(r^6) (dipole-dipole interaction), wouldn't the force be something involving the unit vectors of the vectors connecting the dipole being studied and the ring of dipoles?

More to the point, wouldn't it be the integral per portion of dipoles in the ring dq with some function based off of the distance from the dipoles(charge analogy), so the integral of (kdq/(x^2 + y^2)^(6/2)) cosθ = integral of (kxdq/(x^2 + y^2)^5/2)?

This might help illustrate what I'm talking about: http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html

 

[Vailanor]

. I'm not sure why the book doesn't take this into account, as it would lead to a much more accurate description of the force produced by the ring in comparison to the force produced when the dipoles are pointing up.