zole perfectly definite in its properties, but capable of passing into the latter hydrocarbon by treatment with such powerful reagents as nitric and nitro-sulphuric acids. The existence of at least two distinct yet very similar sulphovinic acids, should not be forgotten; and, as I have shown* some time ago, these acids have their parallels in the methyle series. The derivation of these isomeric acids can, however, be represented by rational formula; while the transformations of turpentine, styrole, and other similar bodies are still obscure. And an observation which I have made throws, I believe, some light upon these metamorphoses. Cymole prepared from oil of cumin and purified by distillation from sodium, boils, according to my determination, at 170°-7; while camphogene, an isomer of cymole obtained from camphor by the action of fused chloride of zinc, boils at about 175°. This isomer of cymole has, however, an odour very different from that of the natural cymole. But Gerhardt, in his treatise on Organic Chemistry, has the following remark :— "Le cymène qu'on obtient par la métamorphose du camphre ne présente pas l'odeur citronnée du cymène naturel; mais on peut communiquer à ce dernier la même odeur qu'au cymène factice, en le traitant à chaud par l'acide sulphurique concentré, et en l'en séparant de nouveau par l'addition de l'eau." I have, in fact, found that natural cymole, boiling at 170°7, passes, by treatment with sulphuric acid, into an oil undistinguishable in any respect from camphogene, the artificial cymole, and boiling, like it, at 175° or 176°.

The circumstances in which the remarkable metamorphoses of cymole, &c. are effected, and the peculiarities of the products, seem to point to the possible equivalency of isomerism among compounds to allotropy among elements, in many instances at least,

Lincoln College, Oxford,
May 1857.

LVII, Note on the Summation of a certain Factorial Expression. By A. CAYLEY, Esq.t

MR.

R. KIRKMAN some months ago communicated to me a formula for the double summation of a factorial expression, to which formula he had been led by his researches on the partition of polygons. The formula in a slightly altered form is as follows; viz.

*Phil. Mag. July 1855, and January 1856.
+ Communicated by the Author.

Σ.Σ

[x+y+2]" [x]" [r+k−x−y]*~1−y [r—1—x]*-1-v
[y+1]"+1 [y]" [k—y]k-y
2k [r+k+2] [r]′

r+3 [k+1]*+1 [k]

[k—1—y]*—1—y

the summation extending from x=0 to x=r-1, and y=0 to y=k-1. In the particular case when k=r, then all the terms of the series except those in which y=x vanish; and putting therefore k=r and y=x, and making a slight change in the form of the right-hand side, the formula becomes

Σ

[2x+2]* [2r−2x] ̃▬1-*

[x+1]*+1 [r-x] ̃ ̄*

= 4.

[2r+1]"-"
[r-1]-

the summation extending from x=0 to x=r−1.

We have in the notation of Gauss [m]"=m.m-1...2.1=IIm, and a factorial [m]" is expressed in terms of the function II by the formula [m]"”=IIm÷II(m—n). Write also

we have

П12m=22mПmП ̧(m—})

II(2m+1)=22m+1IImII,(m+}).

And transforming the factorials by these formulæ, the series becomes

Σ

=

II1(x+1)II, (r− x − }) ___ 2rII, (r+ }) II(x+2)II(r-x+1) II(r+3) the summation, as before, from a=0 to x=r-1. This written

=

This may be

‚II,(x+1). II,(r− x — - 1/1) П2 II (r+1) 8r (r+1) П1(3) II1(r−1) II(x+2) II(r−x+1) ̄ ̄(r+2)(r+3)' the summation from x=0 to x=r-1. The general term does not vanish for a=r or x=r+1, but it vanishes for all greater values of x; hence if we add to the right-hand side the two terms corresponding to x=r and a=r+1, the summation may be extended from x=0 to x=r+1, or what is the same thing, from x=0 indefinitely. The two terms in question are

П2

Σ II, (x + }) II (r− x − 1) __ 112

II(r+1)

4r (r+1)

II() II1(r−↓) II (x+2) II (r−x+1)=(r+2) (r+3)'

the summation from x=0 indefinitely; or substituting for the

functions II and II, their values, the formula is

3.(r+1)

1+

+

+

..(+1)r(r-1)

3.(r-)3.4(r-4) (r−3) 3.4.5(r−})(r—§) (r—§)

+ &c. =

.(r+1)r

4r(r+1)

(r+2)(r+3)'

which is a formula obviously belonging to the theory of the hy

pergeometric series

F(α, B, y, x)=1+2. B

α.β a.a+1.B.B+1
1.γ 1.2.y.y+1

x +

x2+ &c.;

but the formula applicable to the case in hand has probably not been given. It may be proved as follows, premising that I disregard all difficulties arising from infinite values of the functions in the definite integrals, convergency, &c. We have

and hence multiplying by da and integrating from x=0, and again multiplying by da and integrating from x=0 to x=1, we find

+(B+2) d0.0°-2(1—0)—a—y—1

(8+1)(B+2)II (a− 1) II(—a—~—1) 1S,
II(y-1)

if for shortness

Substituting for the definite integrals their values,

П(α-3)II (—a—y+B+1) __ II(a—3)II (—a—y—1)

whence

{(B+1)(B+2)S=

II(a-3) II(-a-y+B+1) I(y-1)
II(a—1) ́ ́ II(—a—y—1) II(~y+ß—1)
II(α-3) II(y-1)

-

1

(α-1)(α-2)
(− 1) (a− 2) (y + 1) (y + 2) — (B+ 2) — _—_ 1 (y.+ 1),

a

(aß+2a-2B+y−2).

For the reduction of the first term we have

II(—a−y+B+1)= [B+1—a—y]®+2 II (~α—~—1),

[B+1—a—y]®+3 — (y+1)(aß+2a−2B+y−2);
[B-y-1]

which is the formula in hypergeometric series required for the present purpose, and which is certainly true when the series is finite.

Write now

a=3, B=r+1, y=r−1;

then the first term is [1]+2÷[1]+1, which vanishes on account of the numerator, and the second term is -r(r+1), and we have consequently

−r(r+2)(r+3)} . }S=~\r(r+}),

which gives

S being here the series in r, the sum of which was required, and the particular case of Mr. Kirkman's formula is thus verified. It

is probable that the general case might be treated in an analogous manner by first grouping together the terms which correspond to a given difference xy, and ultimately summing the sums of these partial series; but I have not examined this question.

2 Stone Buildings, W.C. April 18, 1857.

LVIII. Note on a Theorem relating to the Rectangular Hyperbola.. By A. CAYLEY, Esq.*

THE HE following theorem is given in a slightly different form by Brianchon and Poncelet, Gergonne, vol. xi. p. 205, viz. Any conic whatever which passes through the three angles of a triangle and the point of intersection of the perpendiculars let fall from the angles of the triangle upon the opposite sides is a rectangular hyperbola. And there is an elegant demonstration depending on the properties of the inscribed hexagon. The theorem is, however, a particular case of the following: viz. “Any conic whatever which passes through the four points of intersection of two rectangular hyperbolas is a rectangular hyperbola." And this, again, is a particular case of the following: viz. If there be a conic and a line P, then considering any two conics U, V such that the points of intersection of P, U are harmonics in respect to the points of intersection of P, Q, and the points of P, V are also harmonics in respect to the points of intersection of P, Q, then any conic whatever W which passes through the four points of intersection of U, V will have the like property, viz. the points of intersection of P, W will be harmonics in respect of the points of intersection of P, ; a theorem which is an immediate consequence of the theorem that three conics which intersect in the same four points are intersected by any line whatever in six points which are in involution.

2 Stone Buildings, W.C.

April 23, 1857.

LIX. On the Natural Groupings of the Elements. By WILLIAM ODLING, M.B. Lond., Professor of Practical Chemistry, Guy's Hospital; Secretary to the Chemical Society*.

PART I.

THAT certain elements have certain properties in common is now a time-honoured doctrine in chemical science; but the majority of chemists have been satisfied with a simple admission of the fact: they have not investigated the extent of * Communicated by the Author.