We obtain and As a illustration of our results we shall consider the problem of finding the heat capacity of our strongly cor- related system in normal state and at the low tempera- tures. The heat capacity at constant volume V is equal to By summing these equations we obtain: where the entropy S is given by The quantity µ is the chemical potential at temperature T . On the base of definition of the functional Y ′ Fig. 10 and equation (20) we have At low temperature we may expand F ( µ, V, T ) and µ in even powers of the temperature: As a result we obtain the stationarity property of the functional Y : where µ$_{0}$ is the value of chemical potential for correlated system at T = 0. As a result we have the linear dependence of the heat capacity of the tem- perature at low its values. Therefore to evaluate the co- efficient γ ( µ$_{0}$, V ) it is necessary to obtain the expansion of F in powers of T by using the expression (31), (19) and Fig.10 for Y ′ obtained by us in previous part of the paper. Now we shall discuss the derivative over interaction constant V of functional Y . We shall taken into account the stationarity Y about ˆ Σ and ˆ G c and equation (21). We obtain: We know that the expression (19) for Y and (31) for thermodynamic potential is stationary with respect to changes in the proper self-energy Σ$_{k}$( iω$_{0}$ ) . Therefore be- cause we are interested in the first corrections to F , we can neglect the explicit temperature dependence of mass operator Σ( k | iω ) and G $^{c}$( k | iω ) and replace them by val- ues Σ( k | iω$_{0}$ ) | T =0 and G $^{c}$( k iω$_{0}$ ) | T =0 calculated at T = 0. Thus the first correction to the T = 0 value of F (31) comes only from the difference between the ω$_{n}$ sums in expression (31) and what we would get if we replace them by integrals according to the equation From equation (18) we have: and as a consequence we establish with the solution This constant is F$_{0}$ . Therefore with the stationary property as in superconducting and in normal states. Now it is necessary to consider the functional Y ′ on Fig.10. Since each line of a skeleton diagram of func- tional Y ′ contains an ω$_{n}$ sum, the total first correction to Y ′ is obtained by correcting the computation in each diagram for a single line and use equation (37) for the other ω$_{n}$ sums of the diagram, finally summing over ev- ery line. 1 β ∑ ω n = 1 2 π ∞ ∫ −∞ dω. (37) δF δ Σ( x, x $^{′}$) = 0 , (33) F = F$_{0}$ + Y, (32) F = Y + const . (31) V dF dV = V dY dV , (30) V dF dV = T r ( ˆ G c ˆ Σ) β , (29) V dY dV = V ∂Y ∂V | Σ + V δY δ Σ ∂ Σ ∂V = V ∂Y ′ ∂V | $_{Σ}$= T r ( ˆ Σ ˆ G $^{c}$) β . (28) δY δ Σ( y, y $^{′}$) = 0 . (27) C$_{V}$ = γ ( µ$_{0}$, V ) T + ... (36) F = F ( µ$_{0}$, V, 0) − 1 2 γ ( µ$_{0}$, V ) T 2 + ..., µ ( T ) = µ$_{0}$ + µ $^{′}$T 2 + ..., δY ′ δ Σ( y, y $^{′}$) = ∑ xx ′ δY ′ δG $^{c}$( x, x $^{′}$) δG $^{c}$( x, x $^{′}$) δ Σ( y, y $^{′}$) = 1 2 β ( ˆ G c ˆ Σ ˆ G $^{c}$)$_{y}$ ′ $_{y}$. (26) δ δ Σ( y, y $^{′}$) ( − 1 2 β ) T r { ln( ˆ G c (0) ˆ Σ − 1) + ˆ G c ˆ Σ } = = − 1 2 β ( ˆ G c ˆ Σ ˆ G $^{c}$)$_{y}$ ′ $_{y}$. (25) S = − T ( ∂F ∂T )$_{µ,V}$ . (35) C$_{V}$ = T ( ∂S ∂T )$_{V}$ , (34) δ δ Σ( y, y $^{′}$) T r ln(1 − ˆ G c (0) ˆ Σ) = − G $^{c}$( y $^{′}$, y ) , δ δ Σ( y, y $^{′}$) T r ( ˆ G c ˆ Σ) = G $^{c}$( y $^{′}$, y ) + ( ˆ G c ˆ Σ ˆ G $^{c}$)$_{y}$ ′ y (24) δG $^{c}$( x, x $^{′}$) δ Σ( y, y $^{′}$) = G $^{c}$( x, y ) G $^{c}$( y $^{′}$, x$^{′}$) , (23) V. HEAT CAPACITY [25] 10