^{page 181 note 1} Whittaker, and Watson, , “Modern Analysis” (1920), § 18.3.Google Scholar

^{page 181 note 2} Some properties of this function have been investigated in earlier papers, particularly (i) Proc. London Math. Soc., 31 (1930), 225 ; (ii) ibid., 32 (1931), 87. These will be referred to as papers 1 and 2 respectively.

^{page 182 note 1} ∂ω∂τ is in general a function of τ as well as of the x's, so τ must be eliminated in order that the solution should be expressed as a function of the x's only.

^{page 182 note 2} See, for example, Veblen, “Invariants of Quadratic Differential Forms” Camb.). Math. Tract. No. 24) (1927), 95.Google Scholar

^{page 183 note 1} This in fact follows at once from equations (1) and (10) of paper 2.

^{page 183 note 2} Paper 1, § 2, where ω denotes twice the function here represented by ω.

^{page 186 note 1} See Bateman, , “Electrical and Optical Wave Motion” (1915), 115.Google Scholar

^{page 187 note 1} Since u is an arbitrary constant, the function ƒ of the two arguments and u, is (regarded as a function of x, y, z), an arbitrary function of the former argument only ; that is, of ∂ω∂τ only.

^{page 188 note 1} Whittaker, and Watson, , loc. cit.Google Scholar