In Part II, it was reported that, when smouldering combustion propagates downward through a vertical rod of radius γ with a constant velocity υ, the distribution of temperature T inside the rod is given by
k(∂²T/∂x²)+cρυ(∂T/∂x)+2(1/γ+Δ/γ²)((q/T_i-T_a)-h)(T-T_a)=0    (1)
when referred to an ordinate Ox moving downward with the propagation. The constants k, c, ρ are respectively the thermal conductivity, the specific heat and the density of the rod ; Δ the thickness of a stagnant boundary layer assumed to exist around the rod ; T_i the ignition temperature ; T_a the ambient temperature ; h the heat transfer coefficient ; and q is a constant depending upon the chemical composition of the rod and the ambient air.
The solution of the equation(1) is, in general
T - T_a=θ=e^-αx(Ae^-βx+Be^βx)    (2)
where
α=cρυ/2k, β=1/2k √c²ρ²υ²-8k(1/γ+Δ/γ²)((q/T_i-T_a)-h)
and A and B are integration constants. Taking the fire front as the origin of Ox, the boundary condition is
θ=0,   ∂θ/∂x=0    at  x=0    (3)
As was discussed in Part I, there is no solution of (1) which satisfies (3) in a strict sense, but (3) is fulfilled approximately when
β=0    (4)
This gives us not only υ in terms of c, ρ, k, etc. as
υ²=8k/c²ρ²((1/γ)+(Δ/γ²))((q/T_i-T_a)-h)    (5)
but T in the form :
T-T_a=e^-αx(C＋Dx)    (6)
where C and D are integration constants.
It was reported in Part II that (5) showed remarkable coincidence with experiments. Now in this report, it will be shown that the equation (6), too, coincides satisfactorily with experimental results showing that our theory has got another strong evidence.

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As was reported in Part I and Part II, the formula derived theoretically for the downward propagation of smouldering through vertical solid fuels made of paper in various forms such as circular or rectangular rods, and rectangular strips showed a very good coincidence with experimental results. In this report, hollow cylinders of paper were taken as solid fuels and were examined whether or not the same way of thinking was applicable to them.
As in the former cases, the dependency of propagation velocity υ upon the inside radius γ of the cylinder was studied. And for the sake of convenience in comparing with the former cases, υ² was plotted against γ. The results were very different from those of the former cases. With tubes of cardboard 0.9mm thick, for example, υ² continued to decrease as γ increased from O, and after attaining to its minimum 0.6×l0^-4cm²/s² at γ=1.5mm, υ² began to increase and attained its maximum of 1.5×l0^-4cm²/s² at about γ=5mm. For further increase of γ, υ² decreased gradually tending to an asymptotic value of 0.6×l0^-4cm²/s².
One of the causes of this complicated relationship was considered to be attributed to the air supply being prevented by the inside wall of the tube, and air was supplied forcibly to its inside. The relation of υ² - γ was changed remarkably, and υ² began with 1.7×10^-4cm²/s² (at γ =0) and gradually decreased until at last came to the same asymptotic value of 0.6×10^-4cm²/s²
υ² in the region of γ< 0.5 was too large to be explained with the same way of thinking. However, this discrepancy was eliminated by taking into consideration heat radiation from the upper red-heated part to a fire front through the inside space of the cylinder. As a result of this heat radiation, the relation between υ² and Ta (ambient temperature) was not as simple as were shown in rods. The formula for υ² in terms of γ, Ta, etc. has been left as a pending problem.

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In Japan, it seems very difficult in near future to realize an ideal city filled with in combustible buildings.
1. Conflagration speed formula for housing is shown by formula (7), and in case of within the ranges of 3min≦x≦15min, and 10m/s≧υ≧0, y₂ (the burned area 〔m²〕) is approximately shown by a simple function of x.
2. Conflagration speed formula for large scaled buildings, such as hospital, school, public bath-house, wood work shop, is given in the formula (8), y₃ (the burned area 〔m²〕 is approximately shown by (0.1165σ+1.350) degree.
Within the ranges of 3min≦x≦15min and 10m/s≧υ≧0, Hence 99.87 percent out of all remain below an equation of the 1.7th degree.
3. Since formula (6) is shown by a quadratic of x, in general formula (6)>(8)>(7) and formula (6)> Fig. 6 (y₄)>y₅> formula. (7).
The ratio y₁/y₃ ranges from 1.2 to 2.4 when n=1～2.0, x=10min, υ=3.0m/s, while the ratio y₁/y₂ ranges 1.5 (75% ; ordinary wood housing+25% ; fire-proof wooden housing) to 3.3 (100% ; ordinary wooden housing), n=1～2.0, x=10min, υ=3.0m/s.
Seeing Fig. 2 (Hishida) and Fig. 5 (Hamada), Hamada’s comes larger than Hishida’s.
Therefore, the ratio y₄/y₂ ranges from 1.83 to 2.74 when δ=0.4～0.6, x=10min, υ=3.0m/s.
Accordingly formula (7) is, if δ=0.6 is assumed in formula (6), equivalent to the expression ⌈2.74% ; ordinary wooden housing +74.6% ; fire-proof one ⌋ and, if δ=0.4 is assumed, equivalent to, ⌈25.4% ; ordinary wooden housing +74.6% ; fire-proof one ⌋.

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Analysis on the process of smoke-evolution and evolution-rate induced by thermal decomposition of respective samples at various temperatures have been pursued in terms of dissymmetry factor (Z-value) measurement in light-scattering technique.
The Z-value of some Organic building materials in each temperature were measured by light-scattering method.
Size distribution of smoke in respective systems has been estimated with the aid of ROYCO Particle Counter concerning plastics and woods. The results agrees with ones obtained from Z-data based on light scattering technique and those results obtained from photographic observation by electron microscope.
Estimation on the smoke-evolution rate induced by the thermal decomposition in the controlled stationary electrical furnace from 350°C t_o 650°C have been pursued in terms of turbidity measurement and 90° angular scattering technique concerning respective samples.
The apparent activation energies concerning the evolution of smoke have been obtained by the Arrhenius-type plots on the evolution rate of smoke (k) v.s (1/T).
The reduced activation energies concerning the evolution of smoke have been obtained from the plots of (ΔH/Po₂) v.s Po₂.
Characterization on the mode of evolution of smoke for organic internal building materials have been pursued successfully in terms of the reduced activation energies for the initial stage of fire in the temperature range from 400°C t_o 550°C and for intermediate stage of fire before and after flash-over above 550°C.

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