L. Observations on the Theory of Equations of the Fifth Degree. By JAMES COCKLE, M.A., F.R.A.S., F.C.P.S., Barristerat-Law, of the Middle Temple*.

THE

HE "Reflections," &c. of Mr. Jerrard (Phil. Mag. Suppl. June 1845, pp. 545-574), starting from the solvable form of De Moivre, throw doubts upon the proposition which it was the design of the argument of Abel to establish. The following investigations are based upon a form more general than the quadrinomials of De Moivre or of Euler, which indeed are but particular cases of the quinomial here employed. In order to facilitate the comparison of Mr. Jerrard's results with mine, I have, as far as my own objects and formulæ would permit, adopted his arrangement and notation.

INTRODUCTION.

1. The elimination of a between

25+A1≈4+A2213 +A9x2+A4x+A5=0

where B are known functions of p and A. 2. The first two equations lead also to

x=Q4+Q3Y+Q2Y2 +¶1y3+¶o¥a, where q are known functions of p and A. 3. The elimination of p between

unsymmetric, and the determination of t, u, v and w cannot be made to flow from the solution of a biquadratic.

4. If, however, one of the quantities t, u, v or w vanish, each of the others is expressible in terms of C'. And, denoting by j any value of p, the identical equalities

jt+j3u+j3v=(j2)3t+j2u+(jo)1v

= ( j3)2t + (j3)*u+j3v=( ja)1t+ (ja)3u+ (j4)2v

show that it is immaterial which one we suppose to be zero. Let it be w that vanishes, and let its evanescence change into z and C' into C.

* Communicated by the Author.

5. How, under these circumstances, to express t, u, v and z in terms of C I have elsewhere shown (Phil. Mag. August 1856, p. 124; Diary for 1857, pp. 77, 78). My present object is to inquire whether P1 and P2 can be so determined as to render the equation in y identical with

and, consequently, solvable. Three questions are here involved: (1) the degree of the final equation, (2) its susceptibility of depression, and (3) of solution. The first two are the subjects of this paper.

SECTION I.

6. The identity of y and z is expressed by the system

xx2+P1Xa+P2=t+u+v,
x2+P1x8+P2=it +i^u+i3v,
x2+p1x1+P2=i3t+i3u+i1v,

2

2

ï¿2+P1x¿+P2=ï3t+i2u+iv,
x2+P1xe+P2=i1t+iu +¿3v,

(a)

where i, 2, 3 and 4 are the unreal fifth roots of unity, and a, B, y, 8, e represent in an undetermined or arbitrary order the five suffixes 1, 2, 3, 4, 5, by which the values of x may be distinguished. This system is evolved from

by substituting 2, 3, 4 and 5 (or 1) for i. The right of the last equation is derived from the third form of art. 4 by changing j3 into i, t into v, and v into t. All these changes are permissible, for j may be any fifth root of unity, and t, v and u are as yet wholly undetermined.

7. By combining any four of the equations (a) we can eliminate t, u and v. Replacing x2+P12+P2 by y, we thus arrive at

MaYa+HBY B + μ» Y y + μ¿ Y ¿ + μ« Ye=0,

where μ are functions of i which have no common factor different from 1, and are such that one of them must admit of being equated to zero.

8. These functions admit of five differently derived sets of values, deducible from equations which may be represented by 1μaYa+1μBY B+..+1μe Ye=0,

2 μaya +2μBY B+..+2μ«Ye=0,

5

Sμaya +5 μBY B+.+3μey=0,

and all of which belong to the same system.

9. Since B1(=C1=Σ.y) must be equated to zero, we have

and, eliminating på, there results

Y+=x+2—}Σx2 + (x-−}Σx)p1 ;

consequently, any one of the five equations of art. 8 will enable us to express p1 as a rational function of the roots of the equa

tion in x.

Now, in order that may be different from μZy, λ must admit of being determined so as to make one at least of the v's vanish without any of the others becoming infinite.

First, let

v=0;

then, expressing y in terms of t, u and v, there will arise

among t, u and v.

The last three equations will be satisfied provided

Vg=1+i2, v1=-i-i3 and v——¿3.

Whenever it is necessary to distinguish this from the other corresponding sets of equations, I change v into 'v.

Secondly, let

We need not retrace our steps. The change of i into produces the same effect on the system (a) as the substitution of y, e, B, 8

for B, 7, 8, e respectively; so that if the change be made on the right and the substitutions on the left of the equations involving v (i. e. 'v), those equations will still be satisfied. Hence we may write

2v1 =1+i, 2v=-2-i, 2v=-i and 2=0.

In thus passing from 'v to 2v we have altered the order of the equations (a) without otherwise affecting them. In the new order, however, i fills the place which i held in the old.

Thirdly, let

=0.

The change, in (a), of i into i3 is equivalent to the substitution of 8, B, e, y for B, y, d, e respectively. Hence, proceeding as in the second case, we may write

3v=1+i, 3vß=—¿3—¿a, 3v.=—¿a and 3⁄41⁄2=0. Fourthly, let

В

Change i into ia, and B, y, S, e into e, d, y, ẞ respectively, and proceed in other respects as in the second and third cases. We may write

1v=1+ï3, 1v=-¿a—¿2, ^yy=-22 and 4y=0. Lastly, let λ=-μa, then

-

cannot, so long as the system (a) remains unaltered, conduct to more than one value of p, or,

we may designate it, P.

SECTION II.

12. The substitution in Y of ya, YB, Yy, &c. for ya, Yb, Ye, &c.

respectively is expressed by Y (1⁄4aybye..).

YaYBYY

or, more simply, by

Y(abc..).

αβγ

When the affix of substitution is of the form

(b), it is replaced by (ab),

which indicates an interchange or

transposition of the elements a and b. From the nature of an interchange we have

Y(ab) = x(ba).

13. I premise the following known results of the theory of substitutions :

I. The repetition of an interchange restores the function to its original state. Consequently

II. Let A1, A2 and A, denote any three forms or states which the elements are capable of assuming from changes of arrangement. Then

x(^;)(^;)=x(^;).

The substitutions on the left are called contiguous.

III. A limited number of repetitions of any substitution brings us back to the original function. This number (the degree of the substitution) cannot exceed the number of elements or suffixes contained in the affix when reduced to its most simple form.

IV. Every substitution of the nth degree may be represented by a substitution of the (n-1)th degree followed by an interchange. And any substitution among n quantities may be represented by, at most, n-1 interchanges. This is demonstrated by the introduction of contiguous substitutions. For example, abcd

and

therefore

Y (ader) = Y (abed) (bead) = Y (abc) (cd),

Y (abe)=Y(abc) (bac) = Y(ab) («),

=Y

Y (abed) = x (ab) (10) (cd).

V. It is indifferent in what order independent interchanges are taken.

Thus