“ The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797) ; the form written above is due to Cauchy (1823) . For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. ”

“ Sometimes the upright lines are omitted and a rule of signs is given:—Let the arc s of the curve be measured from some point along the curve in a chosen sense, and let the normal be drawn towards that side to which the curve is concave; if the normal is directed towards the left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned positive, in the contrary case negative. ”

“ Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see Maxima and Minima) . ”

“ If u1 also vanishes, the origin is a double point and u2 = 0 represents the tangents at the origin. If u2 has distinct factors, or is of the form a (y − p1x) (y − p2x) , the value of y on either branch of the curve can be expressed (for points sufficiently near the origin) in a power series, which is either
or
where q1, . . . and q2, . . . are determined without ambiguity. If p1 and p2 are real the two branches have radii of curvature ρ1, ρ2 determined by the formulae
When p1 and p2 are imaginary the origin is the real point of intersection of two imaginary branches. ”

“ Jacobians possess many properties analogous to those of ordinary differential coefficients, for example, the following:—
If n functions (u1, u2, . . . un) of n variables (x1, x2, . . . , xn) are not independent, but are connected by a relation ƒ (u1, u2, . . . un) = 0, then
and, conversely, when this condition is satisfied identically the functions u1, u2 . ”