3D Pendulum Equation

Hey, folks,

it's sunny outside, hope you guys doing well in your workplace or wherever you are.

My question is about the derivation of equation - 3D pendulum

please see this *.pdf below:

Journal paper : "Resonant motions of the three-dimensional elastic pendulum" BY Peter Lynch

http://www.google.co.kr/url?sa=t&source=web&cd=1&ved=0CC0QFjAA&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.77.4110%26rep%3Drep1%26type%3Dpdf&rct=j&q=Resonant%20motions%20of%20the%20three&ei=NAO-Tc6dC4-GvgOSl6G6BQ&usg=AFQjCNEtfq0geuqVEQoNkxWIb56hUk9o0A&sig2=yZVSPm7PjIVWBWh-VPTk6Q&cad=rjt

I tried to uderstand the equation (8) and (9), but couldn't figure out how those equations were derived.

please give me a tip or comment for eq.(8-9)

Thanks in advance.