page 152 note * An implicit assumption is justified in § 22.
page 156 note * These results were first obtained by Clebsch, , Crelle, Bd. lv (1858), p. 269; the preceding analysis differs completely from that used by Clebsch.
page 161 note * It is to be noted that, if they are simultaneous partial differential equations for G, the most general primitive is
where, as regards the two partial differential equations, F denotes the most general arbitrary function of its four arguments.
page 161 note † For another mode of derivation, see § 14, post.
page 163 note * See § 15 (vi).
page 165 note * As x and t are assumed to increase steadily through the range of integration selected, x1 can neither vanish nor become infinite within that range.
page 169 note * It would be that actual resolved part of the displacement of P, if t denoted the arc of the curve ; but, as already stated, it is convenient to select, as the independent variable, a quantity less intrinsic than the length of the arc of the curve which is the main aim of the investigation.
page 176 note * So far as the primitive is concerned, or Q could have a limited number of isolated zeroes when expressed as a function of x alone. But no one of these zeroes could be of an order higher than four, if the integral is to be regular ; and other conditions would have to be fulfilled which here are irrelevant, because Q is devoid of zeroes within the range.
page 179 note * Should C2 be zero, the appropriate selection is to hand, without further calculation.
page 184 note * As special as, e.g., a point of inflexion, or a point of six-point conic contact, on an ordinary plane curve.