page 118 note * The reciprocal of this equation gives the product of a number of factors in H.P., while the product of n factors in O.P. is very easily expressible in terms of exp(n ²) and exp(n).

page 121 note * The symbol ~ means “approximates to a ratio of equality with,” and must not be confused with = or ≈, which mean “approximately equal to.” When followed by a series in ascending or descending powers of x, the words “when x is sufficiently small or large” (respectively) are implied; in other cases the necessary qualifying condition is stated. Unlike the other two symbols, ~ implies that any assignable degree of accuracy can be obtained by taking x sufficiently large or small as the case may be.

page 121 note † It is the essence of this condition that x Is not restricted to being an integer.

page 122 note * When n is large, the factors approximate to equality with 1+x(x−1)/2n², and therefore their logarithms, to x(x − 1)/2²; so the sum of the logarithms is finite inasmuch as the series Σn^-2 is convergent.

page 123 note * It is obvious geometrically from the rectangular hyperbola that

The difference between the first two terms of this inequality is clearly equal to

Expanding and integrating term by term, this integral found to be

But the integral, in the limit, is equal to y, which is thus seen to be equal

page 124 note * If a function is considered only for real values of the variable, to prove that it is analytic in any interval, it is necessary to prove that it possesses derivatives of all orders at all points within the interval; if we are considering the function for complex values of the variable it is only necessary, in order to prove that it is analytic over any region, to prove that it possesses a first derivative at all points of that region, when it can be shown to possess derivatives of all orders at all points within th it region.

page 124 note † t It should be noted that (16) is the interpolation formula that the Calculus of Finite Differences would give for a function satisfying (14). For if, for integral values of x,

page 126 note * Obtainable, e.g. by dividing the equation

by sin θ and making 9→0.

page 126 note † This treatment of the integral in (23) may be rigidly justified as follows : The series for the index in (22) is convergent if v ²< x/2. Let us consider the integral taken between the limits − N and +N where N ¹⁰=x; these limits obviously → ∞ with x, and within this region the index series differs from − v² by less than