As to the theory of age-hardening of quenched alloys, we have two-the distorsion theory and the precipitation theory, but both of them are not satisfactory for the explanation of a large number of observed facts. As the matter of fact, the precipitation can take place, only when the dissolved atoms have locally assembled themselves within the lattice of solvent atoms, thereby causing the distorsion of the atoms and consequently a hardening of the alloys. On the other hand the precipitation itself causes a softening of the alloy, but the presence of precipitated particles in the solvent alloy causes a hardening of it by the slip interference of the precipitates. At room temperature, the result of these three effects will generally be a hardening. but in some cases it may be a softening.
Thus by aging a quenched alloy, the solubility of which for a certain element rapidly increases with the rise of temperature, the hardening is at first to be observed and then a softening. Let H_g be the hardness of the ground mass dissolving a certain element and H_p the hardness increase due to precipitated substance; then the resultant hardeness H is given by H=H_g+H_p
If an alloy be quenched from a high temperature, its hardness increases by the super-saturated solid solution by ΔH_s, but decreases by a certain amount ΔH_p by the diminution of the precipitated substances; hence the quenched alloy has a hardness:
H'=H_g+ΔH_g+H_p-ΔH_p
=H+ΔH_s-ΔH_p
or H'-H=ΔH_s-ΔH_p In the case of alloys. Al-Cu, Al-Mg₂Si, and duralumin, ΔH_s>ΔH_p and therefore H'>H. On the other hand, in the case of alloys Ag-Cu, Mg-Al, the quantity of precipitated substances is large and so ΔH_s<ΔH_p and therefore H'<H. Hence in these alloys, the quenching causes a softening.
During aging at room temperature the alloy becomes hardened by the distorsion of the lattice due to the local concentration of the dissolved atoms before the precipitation, that is by ΔH_dist; at the same time the gradual decomposition of the solid solution decreases the hardness by ΔH_dec and increases it by the presence of precipitated substance, that is, by ΔH_p. Hence the change of hardness of hardness by aging is given by
ΔH=ΔH_dist-ΔH_dec+ΔH_p
Again, by aging at high temperatures, the coagulation of the coagulation of the precipitated particles takes place, resulting in a softening by ΔH_coa, so that
ΔH=ΔH_dist+ΔH_p-(ΔH_dec+ΔH_coa)
The above four terms are all zero initially, that is, immediatly after the quenching, but gradually increase with the lapse of time; ΔH_dist incieases linearly at first and then tends asymptotically towards a certain value. ΔH_p starts a little later, and increases similarly as in the case of ΔH_dist, but in a much less degree. Again, ΔH_dec and ΔH_coa start also later than the hardening of ΔH_dist. Hence the resultant hardness can be obtained from these four curves by summary up the hardnesses corresponding to a definite time. By changing slightly the form of their curves, the resultant curve can take different forms as actually obsserved

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The ternary equilibrium state of the system copper-nickel-silicon relating to liquid was thoroughly investigated by thermal and X-ray analyses as well as microscopic examination, and heterogeneous equilibria were determined by the construction of many sectional diagrams using many specimens.
As shown in Fig. 2 (quantitatively drawn) and Fig. 3 (qualitatively drawn) the diagram is very complicated. The binary systems are not formed on the lines Cu-Ni₂Si, Cu-Ni₅Si₂, Cu-NiSi, Cu₃Si-Ni₂Si, Cu₃Si-Ni₅Si₂ and Cu₃Si-NiSi, the last one being liable to be taken for a binary system but clear microscopically that it is not formed on the line; accordingly no binary system does exist in this system. Moreover, the phases appearing in Cu-Si system do not form continuous solid solutions solutions with those in Ni-Si system.
The significant points in the diagram are as follows:-there exists a ternary compound ω in the composition range of about 12-15% of Si and about 11-12% of Ni, which is formed by the ternary peritectic reaction L+γ+θ→ω at 859°; the γ phase stated in the 1 st. report of this investigation is formed in the ternary system by the binary peritectic reaction L+γ'+γ. the line of this reaction having a peak of temperature at 954° (the point s₃';) and falling from this point to each side rich or poor in silicon.
The reactions and the ranges of 13 primary crystals, 26 monovariant curves and 13 non-variant points are given in Tables 1, 2, and 3.4 respectively.

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