The concept of isotope was a very significant advance in the understanding of radioactivity. But two fundamental questions were still on the table, namely, how do changes in chemical nature occur when an atom undergoes radioactive decay? And what determines whether the atomic number Z increases or decreases in a given radioactive transformation?
In 1913, Soddy in England and Kazimierz Fajans  in Germany answered these questions independently. Both proposed two rules that systematized all the relevant observations of natural radioactivity. They are called Fajans-Soddy rules (some call them laws), radioactivity transformation rules or radioactive displacement rules.
By 1913 the Rutherford nuclear model of the atom was generally accepted. Using this model, a radioactive atom could be considered to have an unstable nucleus that emits an alpha particle or a beta particle (sometimes emitting a gamma ray). Each nucleus has a positive charge given by Ze where Z is the atomic number and e is the magnitude of the charge on an electron. The nucleus is surrounded by Z electrons that make the atom as a whole electrically neutral and determine the chemical behavior of the atom.
On the other hand, we know that an alpha particle has a mass atomic of approximately four units and a positive charge of two units, +2 and . A beta particle has a negative charge of one unit, -e, and very little mass compared to an alpha particle.
With this information in mind, the radioactive transformation rules  say the following:
1 . When a nucleus emits an alpha particle, the mass of the atom decreases by four atomic mass units and the atomic number Z of the nucleus decreases by two units; the resulting atom belongs to an element two spaces back in the periodic table.
2. When a nucleus emits a beta particle, the mass of the atom changes very little, but the atomic number Z increases by one unit; the resulting atom belongs to an element one space forward in the periodic table.
3. When only one gamma ray is emitted, there is no change in the number corresponding to the atomic mass, nor in the atomic number.
These rules, now using the Rutherford-Bohr-Sommerfeld model of the atom help to explain why a change in chemical nature occurs as a result of the emission of an alpha or beta particle. The emission of an alpha particle requires two positive charges from the nucleus and four atomic mass units from the atom. An example is the following:
218 84 Po → 214 82 Pb + α
The new resulting atom ( 82 Pb) with its less positive nucleus can contain two fewer electrons in its outer shells than before, so the two excess electrons are lost. The chemical behavior of atoms is controlled by the number of electrons; therefore, the new atom acts chemically as an atom of an element with an atomic number two units less than that of the original atom.
On the other hand, in the case of beta emission, the nucleus, and with it all the atom acquires a positive charge. An example is the following:
234 90 Th → 234 91 Pa + β
The number of electrons that the atom can contain around the nucleus has increased by one. After it has collected an extra electron to become neutral again, the atom acts chemically like an atom with an atomic number one unit greater than that of the atom before the radioactive change occurred.
Using these transformation rules, Soddy and Fajans were able to determine the place on the periodic table for each of the substances (or nuclides) in the radioactive series; no revision of the existing periodic table was necessary. It is now known that many of the elements between Z = 82 (lead) and Z = 92 (uranium) each contain several isotopes. These results could be deduced from the hypothesis of the existence of isotopes, but direct and independent evidence was also sought and obtained in 1914.
 Some of those forgotten scientists of the first half of the twentieth century, those who passed the Nobel Prize and who would have deserved it. Today his discoveries are in physics and chemistry textbooks, often anonymously.
 At this point in the series, the attentive reader may find the rules worthy of Pero Grullo, but in 1913 they were a breakthrough.