Seshu-Shinno-ke (the hereditary Imperial princes family) (世襲親王家)
The Seshu-Shinno-ke is also known as "the Tei-Shinno-ke" (the legitimate Imperial prince's family). In the Edo period, the Seshu-Shinno-ke was the name for the four princely houses that retained the title of Imperial Prince for generations by order of the Emperor regardless of their genealogical distance from the reigning Emperor.
Summary
From the middle ages, there were several princely houses, including the Yotsutsujinomiya and the Iwakuranomiya, which were not the Seshu-Shinno-ke, but Seshu-Miya-ke (hereditary princely houses) because they did not always succeed to the title of Imperial Prince. These Seshu-Miya-ke had a legitimate right of succession to the Imperial Throne, and were not there only to ensure succession in the event that the Imperial line should die out.
When the Emperor Gohanazono from the Fushiminomiya family acceded to the throne, the bloodline of the Fushiminomiya family became the main line, and the family could have chosen to eliminate the demarcation between the Fushiminomiya family and the reigning Imperial Family so that the Emperor Gohanazono would be succeeded by Imperial Prince Sadatsune FUSHIMINOMIYA, the Emperor Gohanazono's brother. However, Imperial Prince Sadatsune became the head of the Fushiminomiya family without combining its property with that of the Imperial Family, with the Emperor Gohanazono granting the family the right "to call itself the Imperial palace of Fushimi on a permanent basis." This was the beginning of the Seshu-Shinno-ke which could offer a pool of potential successors to the Imperial Throne.
The Seshu-Shinno-ke often contributed to maintaining an unbroken Imperial lineage by providing a successor to the Imperial Throne when a male heir was not present in the direct line of the reigning Emperor. However, after the Meiji Restoration, an Imperial Family Law was passed stipulating a system of permanent Imperial Family membership, and abolishing the Imperial order that had conferred the title of Imperial Prince and the system of the Seshu-Shinno-ke. This was necessary for the resilience of the Meiji system - if the rights of the Seshu-Shinno-ke were accepted under the law, they could not have been easily rescinded, even as the blood line diverged from the reigning imperial line over the generations.
The Seshu-Shinno-ke (Four Imperial prince's families)
The Fushiminomiya family, founded by Imperial Prince Yoshihito FUSHIMINOMIYA, the son of the Emperor Suko of the third Northern Court
In 1947, the family was reduced to commoner status along with 10 other princely houses of the same line.
The Katsuranomiya family, founded by Imperial Prince Toshihito HACHIJONOMIYA, the grandson of the hundred-sixth Emperor Ogimachi
In 1881, the family died out with the death of Imperial Princess Sumiko, the twelfth head of the family.
Note that Imperial Princess Sumiko is the only example of an Imperial Princess (or female member of the Imperial Family) becoming the head of the Seshu-Shinno-ke or a modern princely house.
The Arisugawanomiya family, founded by Imperial Prince Yoshihito TAKAMATSUNOMIYA, the son of the hundred-seventh Emperor Goyozei. In 1913, the family died out, but Imperial Prince Nobuhito TAKAMATSUNOMIYA, the third son of the Emperor Taisho, carried on the religious services of the Arisugawanomiya family.
The Kaninnomiya family, founded by Imperial Prince Naohito KANINNOMIYA, the son of the hundred and thirteenth Emperor Higashiyama. At the end of the Edo period, there was no one to become head of the family, and so a prince of the Fushiminomiya family was accepted as the head of the family. In 1947, the family lost its right to membership of the Imperial Family, and died out in 1988.
Emperors from the Seshu-Shinno-ke
The hundred and second Emperor Gohanazono: the first son of Imperial Prince Sadafusa FUSHIMINOMIYA
The hundred and eleventh Emperor Gosai: the head of the Arisugawanomiya family
The hundred nineteenth Emperor Kokaku: the sixth son of Imperial Prince Sukehito KANINNOMIYA