This is an excerpt from the page 6.18 of book "Properties of gases and liquids, 5th ed". I can figure out the difference between the first one C_pL with other two, but cannot distinguish latter two, C_sigmaL and C_satL. So I fail to derive the rightmost expression in equation (6-6.1). I also looked for other references [1-2] talking about specific liquid heat capacities near the critical point. But none of them makes distinguish among these three definitions of specific liquid heat capacities.
Ref: 1)Hall, W. B. "Heat transfer near the critical point." In Advances in Heat Transfer, vol. 7, pp. 1-86. Elsevier, 1971. 2)Hendricks, Robert C., Robert J. Simoneau, and Ray V. Smith. "Survey of heat transfer to near-critical fluids." In Advances in Cryogenic Engineering, pp. 197-237. Springer, Boston, MA, 1995.