[4] Let the equation V=0 have r equal impossible roots.

the constants K1, L1, K2, L2, &c. may be obtained from the

19

quantities U, U1, &c. as in the preceding case.

The resolution may in all these cases be effected by the method of indeterminate coefficients, by clearing the equation of fractions, and equating like powers of a; this method is however usually much more laborious.

(L. C. D. 380—2; Hirsch, Int. Tab.)

posed may be obtained from the preceding by putting r=0.

[4] R {x, (a + bx + cx2)}} ;

Assume (a + bx + cx2)=c(x+y)2.

[5] R{x, (a + bx-cx2)}};

let (a + bx — cx2)1 = { c(x − e ̧‚) (e, − x) } *,

Assume (a + bx- cx2) = (x — e,) cy.

[6] R{x, (a+bx)3, (a,+b1x)}};
Assume a + bx=(a1+b1x)y2.

-1

[7] _xm-1R {xm, æ”, (a + bx”)}}:

Assume a + bx"=y",

m

which renders the function rational if - is a positive integer.

n

m

Ρ

ax-"+b=y', if is a negative integer.

+

n q

[9] xm-1R {x", (a + bx2 + cœ2n)‡} ;

m

may be reduced to [5] by assuming "y, if is a positive integer.

DD

n

[10] - R{x", (a+b2x2n)}, [bx2 + (a + b2x21)}]} ;

-1

Assume yba+ (a + b2x2n)b.

an−1 R { xmn, (a+bx”)13, (a+ba")1, &c.};
Assume (a+bx")='y¶8.....

(6.) The preceding are particular cases of the following more general forms, in which R1, R., &c. denote any rational functions, and R1, R,1, &c. the corresponding inverse functions.

2

[11] R{x, R'(x)} ;

Assume R1(x) = v.

[12] R{x, R1(x), R21R ̧1(x), &c. R1R... R1(x)}; Assume R1R... R1(x) = v.

1

[18]_R{x, R ̧1(x), R ̃ ̄1(x)]", R ̃ ̄'(x)]2, &c.} ;

Ꭱ

Assume R1(x)=ymn...

[14] d ̧R2(x).{R2(x), R ̃ ̄1R2(x)} ;

Assume R2(x) = v.

[15] R{x, RR2(x)} ;

2

this formula may be rendered rational if we can determine

R1R2(x)=R2(x). R21(x).

[16] dR().R {R, RR, Rmx};

this formula may be rendered rational under the same condition as the preceding.

[17] R{x, R'(x), þ1R ̃ ̄1(x), ¢ ̧R1'(x), &c.}

may be reduced to R {v, P1(v), (v), &c.} by assuming

R11(x)=v.

(Bromhead, Phil. Trans. 1816.)

INTEGRATION OF am-1(a+bx")".

(7.) Let a + bx" = X.

m

If is a positive integer, for a substitute its value in terms

n

of X obtained from the above equation.

m

If+p is a negative integer, assume ax-"+b=Z, and

n

substitute for a its value in terms of Z.

If neither of these conditions is fulfilled, the indices of a and X may be reduced by some of the following formula: from which however no result can be obtained if any of the coefficients become infinite. A result may in these cases be obtained by substituting in the given formulæ for a, b, m, &c. the values which render the coefficients infinite.