Demonstration.-If we denote by x, x2 x3 x4x, the five roots of the proposed equation of the fifth degree, and put, as is permitted,

39

ƒ (x)) = h12+ q x1, ƒ (x2) = h2+q ±2, ƒ (43) = ha+Q 23" } (9.) f(x4) = h4 + q x49 f(x5) = 9x59

and

Q+ q = Q',

(8.)

the result of the elimination of a between the two equations (55.) and (2.) may be denoted thus,

(y—Q'x1—h1) (y—Q'x,—h2) (y—Q'x3—h ̧) (y—Q′ x4—hs). × (y-Q'x5) = 0;

....

(10.)

and if this result is to be of the form (56.), independently of Q, and therefore also of Q', we must have the six following relations :

X1 X2 X3+X2 X3 X4+X3 X4 X5+X4 X5 X1 + X5 X1 X2 +X1 X3 X4+X2 X4 X5+X3 X5 X1+X4 X1 X2+X5 X2 X3 = 0,

h1+h2+h+h1 = 0,

(13.)

(16.)

h1 (X2 X3+X2 X4+X2X5+X3X4+X3X5+X4 X5)

+ h2 (&c.) +hg (&c.)+h1 (&c.) = 0,

.....

(63.)

(64.)

h2 h2 (X3+X+X5)+h, hg (&c.) +h, h1 (&c.)
+h2h ̧(&c.)+h2h4 (&c.) + h3 h4 (&c.)

= 0,

h1 h2 hз + h1 h2 h+h ̧ h3h4+ h2 h3 hş = 0;

3

of which the two first give

and the three last may, by attending to the first and third, and by eliminating h4, be written thus:

h1 (x2-x)+h2 (x22 — x2)+hz (x2—x42) = 0,

2

3

h22 x1+h22x2+h32 x3+ (h1 + h2+hg)2 x4 = 0,
(h2+h ̧) (h ̧+h,) (h1 +h2) = 0.

(20.)

(21.)

(22.)

Selecting, as we are at liberty to do, the first of the three factors of (22.), namely,

and eliminating h, by this, we reduce the two conditions (20.) and (21.) to the two following:

h ̧2(x1+x4){(x,+X4) (X1−X4)2 + (x2+X3)(X2−X3)2 } = 0. (26.)

And from these equations (of which some occurred in the investigation of the former theorem, but are now for greater clearness repeated,) we see that we must have

f(x) = 9x1, ƒ(x2) = 9x2, ƒ(X3) = 9X39
9x19
f(x4) = 9x49 f(x5) = 9x59 ......

(29.)

(45.)

(30.)

(x1+x4) (x,−x4)2+(Xq+X3) (X2−X3)2 = 0,

or at least some one of those other relations into which (65.) and (30.) may be changed, by changing the arrangement of the roots of the proposed equation of the fifth degree.

The alternative (65.), combined with (57.), gives evidently

E = 0;

(60.) but the meaning of the alternative (30.) is a little less easy to examine, now that we do not suppose the coefficient В to vanish, as we did in the investigation of the former theorem. However, the following process is tolerably simple. We may conceive that X1 X2 X3 X4 are roots of a certain biquadratic equation,

x2 + a x3 + bx2 + c x + d = 0,

(66.)

and may express, by means of its coefficients a b c d, the symmetric functions of x, x2 x3x4 which enter into the development of the product formed by multiplying together the condition (30.), and these two other similar conditions,

(X1+X3) (X1−X3)2+(X2+X4) (X2−X4)2 = 0,
(x1+x2) (X,−X2)2 + (X3+X4) (X3—X4)2 = 0.

If we put, for abridgement,

x3+x23+xz2+x43 = f,

...........

{X1 X3 (X1+X3)+X2 X4 (X2+X4)}

{x, x4 (x1+x4)+X2 X3 (X2+X3) }

the condition (68.) will become

f+ g = 0,

...

(67.)

...

(68.)

(69.)

and the product of the two other conditions (67.) and (66.) will become

f2 + hf + i = 0,

so that the product of all the three conditions becomes f3 + (g + h) f2 + (g h + i) f + gi = 0;

and the symmetric functions f, g+h, gh+i, gi, expressed as follows,

gi=a5d-4a3bd+4a2cd+ab c2-c3.

Again, the proposed equation of the fifth degree,

25+ Ba3 + D x + E = 0,

(74.)

(75.)

may be

(76.)

(77.)

(78.)

(79.)

(58.)

must be exactly divisible by the biquadratic equation (66.), because all the roots of the latter are also roots of the former; and therefore we must have

B = b-a2, D=d-a2 b, E = -ad,

and

to the following,

(80.)

(81.)

This relation cab reduces the expressions (76.)... (79.)

f = - a3,
g+h2ab,

gh+i = a+b+ a2 b2 — 4 a2 d,

(82.)

gi = a3 d;

and thereby reduces the condition (75.), that is, the product of the three conditions (66.) (67.) (68.), to the form

Thus, when we set aside these two particular cases, we see by (45.), that under the circumstances supposed in the enunciation of the theorem, the function f(x)-qx vanishes, for every value of a which makes the polynome +Bx3+Dx+E vanish; and that therefore if we set aside the third and only remaining case of exception, namely, the case in which the

x

proposed equation of the fifth degree has two equal roots, and in which consequently the condition (62.) is satisfied, the function f(x) must be of the form (59.); which was the thing to be proved.

Corollary.-Setting aside the three excepted cases (60.) (61.) (62.), the coefficients of the equation (50.) of the fifth degree in y will be expressed as follows,

B'Q' B, D' = Q'4 D, E' = Q' E; ... 5 (86.) and if we attempt to reduce it to De Moivre's solvible form,

so that the relation between y and x reduces itself to the form

.....

y = (x+Bx3+Dx + E) . 4 (x), (87.) which can give no assistance towards resolving the proposed equation (58.) of the fifth degree in .

Observatory, Dublin, June 11, 1836.

IX. Observations on the Fossil Genera Pseudammonites and Ichthyosiagonites of the Solenhofen Limestone, contained in a Letter to R. I. Murchison, Esq., V.P.R.S., &c. By D. E. RÜPPELL, M.D., of Frankfort.*

I

DEAR SIR,

SEND you herewith the few words you requested me to draw up concerning the fossils I wish to exhibit at the Geological Society.

In a paper which I published in 1829, I ventured to express my opinion upon the generic character of two fossils, fragments of which are commonly met with at Solenhofen, and in other calcareous strata of formations of similar age. Having met with some of these fossils in what I considered to be a more than usually perfect condition, I was led to adopt

* Communicated on the part of the Author by Mr. Murchison, who takes this method of laying before the public the notice, which would have been read before the Geological Society, at its last meeting of this session, had not the letter of Dr. Rüppell been missent. The fossils alluded to were exhibited. Mr. Murchison is convinced that this small fragment will be read with interest as coming from the pen of the distinguished traveller whose researches have thrown so much light on the physical geography and natural history of Nubia and Abyssinia.

my present ideas of the animals to which they belonged. I perceived that the forms which naturalists had united in one genus under the name of Trigonellites, Tellinites, Ichthyosiagonites, and Lepadites, (words which are all synonymous,) belonged really to two distinct genera. One of these fossils is not unfrequently found with an Ammonite-like shell, but which has only an apparent likeness to the true Ammonite, for it has no internal septa. In many of these Ammonite-like shells there are found, near to their opening, two calcareous plates resembling in appearance a bivalve shell. These must in my opinion have belonged to the animal which inhabited the Ammonite-like shell, and may have served as a kind of operculum to it, or perhaps as an organ for mastication. I formed this opinion by observing that whenever these two fossils, apparently so dissimilar, are found near one another, the right-hand side of the bivalve-like shell is uniformly of the same length as the largest diameter of the external whorl of the Ammonite-shaped fossil. Besides, I perceived that the two parts which form this kind of operculum are in some other points perfectly distinct in structure from any living bivalve shells, namely, the valves are not connected with ligaments, and have a sharp edge on the side where they unite; the other margin, opposite to this sharp edge, is a thick calcareous mass. The lamina which are on the convex side of these opercula have not, like other bivalves, a central point round which they increase, but are placed somewhat in a diagonal position, a circumstance which is never met with in a real bivalve shell.

Since I wrote the paper alluded to, I have observed a considerable number of these fossils, all of which confirmed the constant proportion of the diameters of the bivalve and ammonite-like shell when found together. I remain, therefore, confident that they belonged to one animal, forming quite a new type in the series of Mollusca, for which I have proposed the name of Pseudammonites.

Some naturalists have expressed the idea that the finding both these fossils so frequently together was a consequence of the animal of the Ammonite-like shell having eaten the other. But if so, how does it happen that there is so constantly a fixed proportion in their relative sizes, and why are more than one pair never found in each shell? Besides, had the one served as food for the other, why are the apparent bivalve shells always in a fine state of preservation, lying parallel to each other?

The other fossil, in shape somewhat similar to these opercula, I consider to be an internal shell met with in a large Third Series. Vol. 9. No. 51. July 1836.

F