^{page 139 note *} When ϕ(x) depends on every value of a function K(x,t) in the range ≤t ≤β is said to be a functional of the function K. In the above equation ϕ is also a function of x in the usual sense.

^{page 141 note *} i.e. that we can find a positive number M such that |K(x,t)| ≤M for all values of x and t in the range considered.

^{page 146 note *} Using the formula if a, β are functions of x. If β, or a, or both, are independent of x, the first, or the second, or both terms, are (respectively) omitted.

^{page 151 note *} See Whittaker and Watson, Modern Analysis, chap. 15.

^{page 158 note *} may be the usual survivance probability, if death is the only cause of exit, or it may be based on a service table, if withdrawals, etc., are also considered.

^{page 160 note *} For, treating the integral as the limit of a sum, we would have p linear equations (corresponding to p values of T in the range 0, T₀) and pq unknown terms l(x_q, t_p) corresponding to the p values of T and the q values of x into which the range (a, b) is divided. There are therefore insufficient equations to determine the unknowns and this remains true when p and q tend to infinity.

^{page 160 note *} The conditions associated with the benefit, as regards number of children, etc., will determine the precise definition of the force i.

^{page 161 note *} For the singular value, λ= 1/A, is greater than 1, A being of the nature of a single premium, allowing for interest, for a contingent benefit of 1.