Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/265

Evidently (B) is the value of (C) when $t=1$. Considering (C) as a function of, we may write

(D)	$f\left(x+ht,y+kt\right)=F\left(t\right),$

which may then be expanded in powers of $t$ by Maclaurin's Theorem, (64), § 145, giving

(E)	$F\left(t\right)=F\left(0\right)+tF'\left(0\right)+{\frac {t^{2}}{2!}}F''\left(0\right)+{\frac {t^{3}}{3}}F'''\left(0\right)+\cdots .$

Let us now express the successive derivatives of $F(t)$ with respect to $t$ in terms of the partial derivatives of $F(t)$ with respect to $x$ and $y$. Let

(F)	$\alpha =x+ht,\qquad \beta =y+kt;$

then by (51), §125,

(G)	$F'\left(t\right)={\frac {\partial F}{\partial \alpha }}{\frac {d\alpha }{dt}}+{\frac {\partial F}{\partial \beta }}{\frac {d\beta }{dt}}$

But from (F),

(H)	${\frac {d\alpha }{dt}}=h\qquad$ and ${\frac {d\mathrm {B} }{dt}}=k;$

and since $F(t)$ is a function of $x$ and $y$ through $\alpha$ and $\beta$,

${\frac {\partial F}{\partial x}}={\frac {\partial F}{\partial \alpha }}{\frac {\partial \alpha }{\partial x}}\qquad$ and ${\frac {\partial F}{\partial y}}={\frac {\partial F}{\partial \beta }}{\frac {\partial \beta }{\partial y}};$

or, since from (F), ${\tfrac {\partial \alpha }{\partial x}}=1$ and ${\tfrac {\partial \beta }{\partial y}}=1$,

(I)	${\frac {\partial F}{\partial x}}={\frac {\partial F}{\partial \alpha }}$ and ${\frac {\partial F}{\partial y}}={\frac {\partial F}{\partial \beta }}.$

Substituting in (G) from (I) and (H),

(J)	$F'\left(t\right)=h{\frac {\partial F}{\partial x}}+k{\frac {\partial F}{\partial y}}.$

Replacing $F(t)$ by $F'(t)$ in (J), we get

$F''\left(t\right)=h{\frac {\partial F'}{\partial x}}+k{\frac {\partial F'}{\partial y}}=h\left\{h{\frac {\partial ^{2}F}{\partial x^{2}}}+k{\frac {\partial ^{2}F}{\partial x\partial y}}\right\}+k\left\{h{\frac {\partial ^{2}F}{\partial x\partial y}}+k{\frac {\partial ^{2}F}{\partial y^{2}}}\right\}.$

(K)	$\therefore F''\left(t\right)=h^{2}{\frac {\partial ^{2}F}{\partial x^{2}}}+2hk{\frac {\partial ^{2}F}{\partial x\partial y}}+k^{2}{\frac {\partial ^{2}F}{\partial y^{2}}}.$

In the same way the third derivative is

(L)	$F'''\left(t\right)=h^{3}{\frac {\partial ^{3}F}{\partial x^{3}}}+3h^{2}k{\frac {\partial ^{3}F}{\partial x^{2}\partial y}}+3hk^{2}{\frac {\partial ^{3}F}{\partial x\partial y^{2}}}+k^{3}{\frac {\partial ^{3}F}{\partial y^{3}}},$

and so on for higher derivatives.