ON INTEGRATION BY RATIONALIZATION. BY BENJAMIN WILLIAMSON, A. M., Fellow of Trinity College, Dublin.

THE following method of showing that the various modes of rationalization of the expression F(x, √ a + 2bx + cx2) dx, (where Fis rational and algebraic), are cases of one general transformation, may be worthy of the notice of the student.

It can be easily seen, as is proved in treatises on the Calculus, that any expression of the foregoing form is reducible to

where L and M are rational functions of x.

Now suppose a and ẞ to be the roots of the equation

-

√c {λ — aλ' + 2(μ − a μ') z + (v — av') z2 } { λ — ßX' + 2(μ — ßμ') z + (v − ßv') za } (1)

This expression obviously becomes rational if the quadratic factors, under the radical sign, be each made a perfect square.

or

This requires

(μ - αμ') = (λ - αλ') (ν - αν' ),

μ3 − dv + (dv' + vλ' − 2μμ') a + (u'2 – λ'v') a2 = 0 ;

'This is an application of the general method of Transformation of Jacobi.

(2)

See Fundamenta Nova Theoria Functi

onum Ellipticarum.

and a similar equation with ẞ instead of a. Moreover, by hypothesis, a satisfies the equation

a + 2ba + ca2 = 0;

accordingly (2) holds, if the constants λ, uμ, &c., satisfy the equations

μ2 - λv = Ka, dv' + vλ′ – 2μμ' = 2Kb, μ22 - X'v' = Kc, (3) where K is any constant.

Again, solving for ≈ from the equation

we have

x (λ' + 2μ'z + v'z2) = λ + 2μz + vz2,

(v − xv') z+μ−xμ'=V/ μ2−dv+(d'v+v'λ − 2 μμ')x+(μ ́2 −λ'v') x2

=√ K (a + 2bx + cx2).

Also, by differentiation, we get

(λ' + 2μ'z + v'z2) dx = 2 {μ + vz − x (μ' + v′z) } dz

√ a + 2bx + cx2 λ' + 2μ'z + v'z2°

Hence we see that any algebraic expression of the form

(4)

(5)

is rendered rational by this substitution, provided λ, μ, &c. are rational quantities for which equations (3) hold.

These equations admit of being satisfied in a number of ways. We proceed to consider the simplest cases :

(1). Let a be positive, and we may assume v = 0, μ' = 0, and K = 1; this gives

Moreover, without loss of generality, we may assume v = 1, which gives λ = 20, λ =- c,

Equation (4) becomes in this case

√ a + 2bx + cx2 = √α- xz,

which agrees with the well-known transformation. (2). Next if we assume v' = o, μ = 0, and v = 1,

It may be observed that since a and ẞ do not enter into these results, they hold whether the roots be real or imaginary.

Again, when the roots are real, we can rationalize the radical in (1) by making one factor reduce to a constant, and the other to g2.

or

Accordingly let

λαλ' =ο, με αμ' = 0, μ- βμ' = 0, v - Bv'ao,

ing, as it is in some respects simpler than that usually employed. Let the roots be arranged in order of magni

tude, i. e. a > ß> y > 8: and assume x =

a

I 22

then, in ac

√ A (x − a) (x − ẞ3) (x − y) (x−8)= √ A(a−y− (B − y) z2) (a− d−(B–d)≈2

it may be denoted by k2; and, on making y = sin 0, the expression reduces to the form

do

LVI-k2 sin20

where

L = √ A (a− y) (3 − 8).

Again, if A be negative, we assume x =

transformed expression is (writing - A instead of 4),

This is of the required normal form, having its modulus the complement of that in the former case.

It may be shown that the ordinary transformation of elliptic integrals of the first species is easily arrived at by the foregoing method.

For, if we rationalize the expression

dx

by the