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y, may be

tial Auxion relative to y, by

i

y the denominators shewing the

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

i ť represented by x +y; where - are the partial fluxional

y

у 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

3

Ty

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i

y, taken relative to y, will be expressed by x or more simply by

y 7 y; and the partial Auxion of!, relative to x, by 7

x: where y

yx ¿ 7 and correspond to P and Q of the preceding example, and

. ху

yox are called partial flucional coefficients of the second order ; the denominator xy denoting that the fluxion is first taken upon the supposition of x alone varying, and then again of y

alone varying; and the denominator yr 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

į :=Pē, if we take the fluents, supposing only z as variable, we shall have 1=Pxta correction which may be some function Y of y, as is evident. Therefore t or S (Px+Qy)=/ Pr +Y. Similarly, integrating on the supposition of y, being the only variable, we have

1=S (Pc+Qy)=S +a function X of x. Therefore SQj+X=Pi+Y, or by transposition,

SQý— Po=Y-X.

ince

Hence if in the quantity SQy-SPc we suppose x=0, we shall obtain the value of Y; and if we suppose y=0, we shall get the value of_X, and by these means determine the integral of Ps+Qy. For example, let the proposed quantity be (3 x? +2 bxy-3y) 2

Ý +(bx*–6 xy +3 cy”) j; which is integrable, because = 2b3

y 6y=R. We shall have spr=r? +by x*—3y 'x, and SQý = br' y - 3 x y + cy". Consequently Y---X = cyl-6; making sro, we obtain Y = cy, and if we suppose c = , we shall have X=?? Therefore the integral of the proposed fluxion is

goz + 6 x 'y3x y* + cy+C. 98. We may find the quantity Y without having recourse to /Qy. For since S(Pr+Qy) = SP: +Y, it is evident that if we take the Auxion of SPč, supposing only y to vary, so that the result may be 'Py, we must have Qy='Py + Ý, therefore Y=S (Q—'P) y. Thes in the preceding example S Pi = x + bx'y – 3 x y', of which the fluxion, supposing only y to vary, is 'Py=(bx*—6 xy) . Therefore Y=S (Q—'P) y=, 3 c yo y=oyo, as before.

Eramples.

j 1. Required the fluent of 2 bx x +

+ay?

✓ (1+y) 2.

yy
♡ (* +y)

(
yx —xy ?

x2 + y2 99. If a fluxion of three variables, as Pr+Qý + R2 be proposed, calling its integral t, we shall bave

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

P

Å the proposed fluxion is compleat, we must have

y Ř

These three conditions being fulfilled, the integral will be y På+V, V being a function of the other two variables y and 2.

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3.

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p' = 2628. Consequently V = {1 (+ zu)

100. To determine it, take the fluxion of SPë, supposing y and s to vary, and we shall have a quantity of the form P'y +P"z; we must therefore have V+P'y+P"ż=Qý+Rz; and consequently V=S {QP)+(R—P)}, an integral in which there enters only two variables, and which may be had by the preceding method. It is evident that we might have found the integral by means of JQy, or Ś R2, in the same manner as by / Pc. For example, let the proposed quantity be (2 x y* +4 bz? x) + y + 3y2 +2y **}} + {42+2 ** 2+

y + which satisfies the three conditions necessary to its being integrable; we shall have sPr=yx2 +bz2 x4, of which the Auxion, taken on the supposition of both - and y being variable, gives P=2yr, " =

=

y {

+ 3y? } ý +

yy {'+

yy + zz (42

3y "
+xz;} :=47i+34 °ý +
v (yy + zz

the integral of

V(yy+zz)' which presents itself immediately without the assistance of the preceding method, and we obtain V=x+y+ (yy+zz). Therefore that of the proposed fluxion is + +y + Vyy+zz) +y* ** + bz* * +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 Pë+Qč?, in which P and Q are any functions of the variable t. If we consider x as a variable y, the proposed Auxion becomes Py + Qyi. Now that this may be integrable, we must have Alus (Q 3) ; but only - and its powers enter into P, and y is

ly y

'P not found in Q. Therefore

V _Qy

=Q, or P = Qi; the condi

473 +

tion necessary that a fuxional expression of the second order Pä+ Qiri may be integrable. If this condition is satisfied, we have S(Pë+Q;-)=S (Pöti P)=s Py =Py=Pz.

3

m

Example. The Auxion mm-1 ö+m (m—1) samt is integrable, because P = m (m2---1)-=Qi; and the integral is mani, which being again integrated, gives ira +C, for the primitive Auent.

102. If i has been supposed constant, the Auxion is Qir?, of which the integral, (since P=S Që), is is Qi + the constant Ců.

For example, sä? (1-TI)=is (-i)==(x-x)+Cř, and integrating again, we obtain Cx+C'+ xr- **.

103. Let there be proposed the general fluxion of the second order Pë+Qr?; if we take its fluxion, we shall obtain Pë+(+2Qi)

P ö+Q?. Therefore reciprocally the general fluxion of the third order Rä+Scë+T 23 will be integrable, or reducible to a fluxion

Ř

S

of the second order, if =STi ; and the fluent will then be

2

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Reti? STC.

For example, x2 ö+2x} *+(3xx-1)*, possesses the condition of integrability, and its integral is x? *7** ().

104. If i be supposed constant, it is then clear, without any condition, that j'T * =* ST++C; the integral of this fluxion is isiSTi) +Cr ++C'*; lastly the integral of this again is sits Sti+ Cogn+Cx+c".

C'rC".

Crx

2

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For example, 1x3=

+ Cx2 +C++C". (m + 1) (m+2) (m+3) In a similar manner we may find the fuents 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+Qý+Rr? +Sry +Tj?. In order to find ihe conditions of the coefficients P, Q, R, &c. take the Auxion of Ar +By, in which A and B are any functions of r and y, and we obtain Ař+ Bij + Ar+ By. Now since A is a function of both r and y,

Å by art. 96, we have A- s+y; consequently the fluxion of

y

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therefore the proposed Auxion is integrable, whenever R = S=P+, and T =

y 106. These conditions being fulfilled, the integral will be Pë+Qy; and if i has been supposed constant, the integral of Qý + Rx++ siry+Ty', will be Qý +ë fR3+Cs, (because P=sRi), and the conditions of this integral will be T=Q, and S=Q_fur (SRzj*

y For example, 6 x2 ¿+6 xy x2 + x3y, in which 3 is constant, possesses the preceding conditions, and its integral is xs y +3x2 yx + Cë; 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.

y

• In this last equation of condition the expression

(SRx)

denotes the partial fiaxional useffi rrent of SRi relative to uj,

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