tial fluxion relative to y, by

y.

y; the denominators shewing the

variable relative to which the fluxion is taken. Consequently the entire fluxion of any function t, of two variable x and y, may be

represented by+y; where,

i

are the partial fluxional

y

coefficients of the first order, and correspond to P and Q in the preceding theorem.

Upon the same principles of notation, the partial fluxion of

xy

y; and the partial fluxion of.

relative to x, by

and correspond to P' and Q' of the preceding example, and

ух

are called partial fluxional coefficients of the second order; the denominator ay denoting that the fluxion is first taken upon the supposition of x alone varying, and then again of y alone varying; and the denominator yx denoting an arrangement exactly the reverse.

97. From the preceding theorem it follows therefore that, the fluxional coefficient of the second order of any function of two variables, taken first with respect to one of these variables, and then with respect to the other, is the same in whatever order the operations are performed. And consequently if any fluxional expression of the form Px+Qy admits of exact integration, the fluxional coefficient of Q taken relative to x, must be equal to the fluxional coefficient of P, taken relative to y.

98. If this condition is fulfilled, the integration is easy. For

i since P, if we take the fluents, supposing only a as variable,

x

we shall have t=ƒfPx+a correction which may be some function Y of y, as is evident. Therefore t or (Pr+Qy)=/P+Y. Similarly, integrating on the supposition of y, being the only variable, we have t=f(Px+Qy)=SQy+a function X of x.

Therefore Qy+X=ƒ Pr+Y, or by transposition,

SQy-fPx=Y-X.

Hence if in the quantity sQy-SPx we suppose x=o, we shall obtain the value of Y; and if we suppose y=o, we shall get the value of-X, and by these means determine the integral of Pr+Qy. For example, let the proposed quantity be (3 x2+2 bxy—3 y3) x

+(b.x2—6 xy+3 cy2) ÿ; which is integrable, because

бу=Q.

y

=2bx

We shall have Pr=x3+by x2-3y 2x, and SQÿ = br2 y

-3 x y2+cy3. Consequently Y-X = cy3-x3; making ro, we obtain Y=cy, and if we suppose ro, we shall have X=r. Therefore the integral of the proposed fluxion is

x2+bx 1y-3x y2+cy3+C.

98. We may find the quantity Y without having recourse to fQy. For since (Pr+Qy) =SPx+Y, it is evident that if we take the fluxion of Pr, supposing only y to vary, so that the result may be 'Py, we must have Qy='Py+ Y, therefore Y=S(Q-P) y. Thus in the preceding example /Px=x3 + bxy - 3 x y2, of which the fluxion, supposing only y to vary, is 'Py=(bx2-6xy) y. There fore Y=(Q-'P) y=f 3 c y2 y=cy3, as before.

99. If a fluxion of three variables, as Pr+Qy+ Rz be proposed,

And it may be shewn in the very same way as in article 95, that if

y

These three conditions being fulfilled, the integral will be

SP+V, V being a function of the other two variables y and z.

100. To determine it, take the fluxion of SPx, supposing y and z to vary, and we shall have a quantity of the form Py+P"%; we must therefore have V+Py+P"z=Qy+Rz, and consequently V=S

{(Q—P') ÿ + (R—P') ż}, an integral in which there enters only j

two variables, and which may be had by the preceding method. It is evident that we might have found the integral by means of Qy, or JRz, in the same manner as by Pr.

For example, let the proposed quantity be (2 x y2+4 bz2 x3) x +

y

{ √(yy + z) + 3y2 + 2y x2 } ÿ + { 42*+26 x* z+

which satisfies the three conditions necessary to its being integrable; we shall have fPx=y2 x2+bz2 x2, of which the fluxion, taken on the supposition of both x and y being variable, gives P'=2y x2,

which presents itself immediately without the assistance of the preceding method, and we obtain V=z++y3 +√/ (yy+zz). Therefore that of the proposed fluxion is 2*+y3+ √(yy+zz) + y2x2+bz2 x2+C. After what has been shewn, it will not be difficult to ascertain the conditions which must be fulfilled in fluxional expressions containing a greater number of variable quantities, and to integrate them when these conditions take place. These equations are called equations of

condition.

101. Thus much premised, we will now proceed to the integration of second fluxions. Let us first take the fluxion of the second order Pr+Q2, in which P and Q are any functions of the variable x. If we consider x as a variable y, the proposed fluxion becomes Py+Qy. Now that this may be integrable, we must have P_flux (Q5); but only and its powers enter into P, and y is

y

not found in Q. Therefore

P_Q

=Q, or PQr; the condi

tion necessary that a fluxional expression of the second order Pr+ Qr may be integrable. If this condition is satisfied, we have S(Pï+Qi2)=ƒ (Pï+i P)=ƒ P¥=Py=Pi.

Example. The fluxion m xTM-1 ä+m (m—1) *”—2 2 is integrable, because Pm (m-1) x-x=Qx; and the integral is mal i, which being again integrated, gives "+C, for the primitive fluent.

102. If x has been supposed constant, the fluxion is Qr2, of which the integral, (since P=f Qx), is x f Qr+the constant Cr.

For example, ƒx2 (1—xx)=x ƒ (x−xx x)=x (x−‡ x3)+Cx, and integrating again, we obtain Ca+C'+} xx—‚1⁄2 x1.

103. Let there be proposed the general fluxion of the second order Pr+Qx2; if we take its fluxion, we shall obtain Pr+(P+2Qx) +Qx2. Therefore reciprocally the general fluxion of the third order R+S+T 23 will be integrable, or reducible to a fluxion

For example, 2x+2x3 x x+(3xx—1) 3, possesses the condition of integrability, and its integral is 2x+x2 (x3—x).

104. If be supposed constant, it is then clear, without any condition, that ƒT3 = x2/Tr+Cr2; the integral of this fluxion is ¿ ƒ (¿ST x) + Cx x +C'; lastly the integral of this again is si fr STi + Cxx+C ́1+C".

2

For example, " x3 ==

2-3

(m + 1) (m + 2) (m+3)

+ 1⁄2 Cx2+Cr+C".

In a similar manner we may find the fluents of higher orders of fluxions, and the conditions of their coefficients.

105. We shall now proceed to the consideration of fluxions of the second order, containing two variable quantities; these may be represented generally by Pr+Qy+Rr2+Sry+Ty2. In order to find the conditions of the coefficients P, Q, R, &c. take the fluxion of Ar+By, in which A and B are any functions of x and y, and we obtain Ax+By+Ar + By. Now since A is a function of both ≈ and y,

therefore the proposed fluxion is integrable, whenever R =

106. These conditions being fulfilled, the integral will be Px+Qy; and if has been supposed constant, the integral of Qy + Rx2+ Say+Ty2, will be Qÿ+*ƒRx+Cx, (because P=ƒRx), and the conditions of this integral will be T=2, and S=Qflux (Rr)*

Q

y

x y

For example, 6a2 àÿ+6 xy x2+x3 ÿ, in which a is constant, possesses the preceding conditions, and its integral is 3 y+3x2 yx+Cx; this result gives by a second integration xy+Cx+C'.

In a similar manner we may investigate the conditions required for a greater number of variables.

• In this last equation of condition the expression

fluxional

efficient of ƒ Rx relative to g

(fRx)

denotes the partial