The following two fundamental proposals as to the definition and the scale of hardness are given: 1. The hardness of a material is its ability of unyieldingness by compression. 2. Let the (σ) strain (e) relation of a material by simple compression be e=f(σ), then the hardness number is H=∫^1_0σde. To find this new hardness scale from the results obtained by the Brinell and the drop ball hardness tests, several formulae which represent the relations between the applied load or energy and the amount of indention, are deduced from several probable assumptions; then the proposed new hardness scale is calculated from those results. As the by-products, many phenomenae which have been already discovered from experiments, are proved mathematically. At first the present author made the following two fundamental proposals as to the definition and scale of hardness; 1. The hardness of a material is its ability of unyieldingness by compression. 2. Let the stress-strain relation of a material by simple compression be e=f(σ), then the hardness number is H=∫^1_0σdε But since the compression test is not proper for the commercial hardness test, the present author has shown the approximate methods of calculation of the above mentioned hardness from the results of the Brinell hardness test and the drop ball hardness test. Assuming, that 1. Frictional resistance between ball and specimen is negligible, 2. Loss of energy due to shock of impact negligible 3. Stress-strain relation by simple compression is e≡ε/1-ε=(ασ)^m and 4. The hollow elementary cone A B C D is compressed by the applied load into A' B' C D in Fig.4, we obtained the formulae 7, 8, 26 and 27, which represent the relation between the amount of indentation or harness number and the applied load P or energy W, and we compared these equations with the existing experimental formulae 9, 28 and 31, and proved theoretically many phenomenae which were found already by experiments. Thus we formed the equations by which we can calculate the value of α and m from the results of the Brinell or drop ball hardness test, which are shown in Eqs.10, 11, 29, 30, 32 and 33. Consequently the proposed new hardness scale H can be calculated by Eq.38, or approximately by Eq.40. In conclusion, the author wishes to express his heartfelt thanks to Prof. C. Sunatani, who has given him very much valued advice and encouragement, and to Mr. E. Nakagawa, who has assisted him in both the very troublesome calculations and the large number of experiments. He also wishes to acknowledge his indebtness to "The Saito Gratitude Foundation" for finnacial aid.

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There have been considered several emperical methods to investigate the stress distribution in bodies subjected to external forces. Among them, the most remarkable and successful one is the photo-elastic method which owes much to Prof. Coker. This method gives much hints to the mathematical theory of elasticity, and on the other hand, emperical estimations of many problems which cannot be theoretically solved. I have made some experiments for the discussion of the following four problems, i.e., (i) The stress distribution in the thin round disc compressed with two parallel planes. (ii) The same as (i) but having a very small hole at the centre. (iii) The change of the distribution of stress when the hole becomes larger and larger. (iv) The accuracy of the theory of curved beams, commonly adopted. For the sake of simplicity, I limited the observations of stress on the horizontal and vertical axes. The stress lines were traced on a quadrant. Conclusions: - (i) The experiment (i) showed a good agreement with the theoretical calculation by Prof. Foppl. (ii) A very small hole at the centre of disc induces a very large amount of stress on the hole boundary in the direction of 45゜with the horizontal axis. (iii) The theory of curved beams, generally adopted, is very doubtful; it gives too large amount of stress in case of thin plate, at least.

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