Question: Manipulating equations; substitutions

I am trying to enter equations from the following journal article: The link can be found in the Download 4865_Page from Lo_ApplMech_1962_V1_pp691-695.pdf
View file details Thus far I have been able to create equation (8) of the article in the attached worksheet: The link can be found in the View 4865_ring on MapleNet or Download 4865_ring
View file details Any suggestions on how to substitute equation (9) of the article and come out with equation (10)? Thank you for your assistance.
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