Note on a functional equation employed by Sir George Stokes

1. In a paper by Sir George Stokes "On the Intensity of the Light Reflected from or Transmitted through a Pile of Plates," the author forms and solves two functional equations, viz.:—Φ (m +n ) =Φ (m ) +Φ (n ) {Ψ (m )}2 /1-Φ (m )Φ (n ), (1)Ψ (m + n ) = Ψ(m ) Ψ(n )/1-Φ (m )Φ (n ). (2) The symbolsm, n represent, as was required by the object of the paper, positive integers; but in the course of the investigation he deals with them as continuous variables, solves the equations (1) and (2) on that footing, and finally verifies the equations whenm, n are integers. The two equations are treated as independent, and the results are thus expressed,Φ (m )/sinmβ =Ψ (m )/sinα = 1/sin (α + mβ ), (3) whereα, β are constants. In what followsm, n will be taken to be ordinary continuous variables.