page 311 note * It may be necessary to explain that a maximum is not necessarily the greater nor a minimum the least value that a function can assume. A function of the nth degree has usually n–1 maxima and minima values; and their characteristics are that a maximum is greater than preceding and following values, and a minimum is less. Thus, if a be a value of x that renders Φ(x) a maximum, we must have Φ(a) greater than both Φ(a + h) and Φ(a – h); and if a renders Φ(x) a minimum we must have Φ(a) less than both Φ(a + h) and Φ(a–h), h being in both cases supposed indefinitely small. It is obvious then, that it is amongst the maxima and minima values of a function that the greatest numerical value it can assume must be sought.

page 311 note † The above is a case of the following more general theorem:—If Φ(x) be a function of the nth degree whose roots are

and if k = ½ (n – 1)t, substitution in Φ(x) of k + a and k – a successively, for x (a being any number, rational or irrational), will give results equal in numerical value, and of the same or different signs, according as n is an even or an odd number. That is, we shall always have, in a function constituted as above,

This theorem, which I do not recollect to have met with anywhere, is very easily proved.

page 313 note * The function which in the Theory of Equations receives the name of the derived function of Φ(n), is that which in the Higher Analysis is known as the differential coefficient of Φ(n).

page 314 note * What we learn from the occurrence of cipher here is that 7, the transforming number, is a root of the quadratic whose coefficients are those of the first three terms of Φ'(n + 3), viz., n²–10x + 21. The other root is obviously 10– 7 = 21÷7, =3 . These however are matters with which we are not particularly concerned here.

page 316 note * This would be apparentfromthefigurewere it correctly drawn, and on a sufficiently large scale. But it is not correctly drawn, the dotted curve being sketched by hand. And the deviations in the figure are not necessarily greatest at the points indicated. Also, from the necessity of the case, the dotted curve is so drawn that the portions of it situated between the points of intersection with the continuous curve are alternately concave and convex to the axis of abscissæ, in consequence of which the points in question become points of contrary flexure. Neither is this necessarily the case. It will only be so when the neglected term in ux+n is comparatively large. It is easy to conceive, though very difficult to draw, a curve which, while lying alternately on opposite sides of the given curve, shall yet, like it, be convex to the axis throughout.

page 318 note * The roots are multiplied by 10 at the outset, the first root figure here being in the first decimal place (32).

page 321 note * These extra places are irrespective of, and in addition to, those arising in consequence of the fractional values of the several differences in (81).