(a) In each city of the second class, subject to an affirmative recommendation of the Pension Board submitted in writing to the council, the council may by resolution permit employes now ineligible to join the Pension Fund to be admitted to the Pension Fund in accordance with this section. Upon the adoption by council of such resolution, this section shall take effect as follows:

(b) Every city or authority employe otherwise eligible to join the Pension Fund created under the act to which this is a supplement but who is now ineligible to join because of not having done so within the times provided by such act, shall have the option of joining such Pension Fund within six months after the effective date of the aforementioned resolution. Any city or authority employe who joins such pension plan shall receive full credit toward his pension under the act for each year of service as an employe of the city or authority, beginning with the date on which he was first eligible to join the Pension Fund upon his producing proof satisfactory of the Pension Board of the number of years of such service, and upon his making back payments at the same rate as if he had been a city employe and a member of the Pension Fund during such period, in a lump sum or by installments as may be determined by the board. If the employe is retired before the back payments have been completed, the annuity shall be reduced by an amount equivalent to the unpaid back payments or deduction increases not anticipated.

(c) Every city or authority employe presently a member of the Pension Fund who has not received full credit toward his pension under the act, to which this is a supplement, for each year of service as an employe of the city or authority, beginning with the date when he was first eligible to join the Pension Fund, may by written election filed within six months after the effective date of the aforementioned resolution, upon producing proof satisfactory to the Pension Board of the number of years of such service, and upon making back payments at the same rate as if he had been a member of the Pension Fund during such period, in a lump sum or by installments, as may be determined by the board, receive full credit toward his pension. If the employe is retired before the back payments have been completed, the annuity shall be reduced by an amount equivalent to the unpaid back payments or deduction increases not anticipated.