point, and then let go, it is evident the less the resistance of the attached body, the greater the ascending arc described by the pendulum, and vice versâ: consequently the greater or less resistance will be measured by the arc of vibration. Two disadvantages attend this mode of experimenting; the slowness and inequality of the motion, and the passage of the figures through the water not being rectilineal.

Subjoined are experiments made in this manner with a pendulum 5 feet 5.85 inches long, the lower extremity being immersed 12-7 inches.

The solids were two inches in diameter, and as much in length, with the exception of the double cone, which was four inches long, when lengthened by a cylinder, it measured six inches (the same remark is applicable to the elliptical spindle); and the sphere, when cut in halves, and separated by a cylinder, measured four inches.

Resistance of a sphere cut in halves, and length

Resistance of the base of a wedge

1000

Resistance of the vortex, its angle being 53.08
Resistance of a double cone

Resistance of the same lengthened by a cylinder.. 380
Resistance of an elliptical spindle

Resistance of an elliptical spindle lengthened by a
cylinder *

1000

735

The following table contains experiments made with six different solids, the diameter of each being two inches, and the length seven inches.

It is remarkable that the simple addition of length should so much diminish the resistance, a circumstance fully corroborated by other experiments made in a different manner.

13

5. A ditto, ditto, greatest breadth from the fore-
most end

...

6. A ditto, ditto, greatest breadth from the fore-
most end

587

540

The experiments were compared with each other in the following manner. The ascending arc of the pendulum, before any body was fixed to it, was found by measuring the chord, and calculating the angle to be 19° 5′ 39′′, the elliptical spindle being attached to the rod, the ascending arc was found to be 18° 42′ 35′′; the difference between those two numbers is 1384". This solid being detached, and the circular spindle substituted, the arc of ascension was found to be 18° 46′ 07′′, which, deducted from 19° 5′ 39′′, leaves 1172. Then 1384 1000 :: 1172: 897, the comparative resistance of the circular spindle. In the same manner the other comparative resistances are calculated. From these experiments it appears that the extreme breadth should be placed from the bow; but whether this will hold good in all velocities remains to be determined,

It was my intention to have made some remarks on the method of cutting sails; but lest I should intrude too much on your time, I beg leave to subscribe myself,

My dear Sir, very sincerely yours,

MARK BEAUFOY.

ARTICLE III.

Elementary Ideas on the First Principles of Integration, by Finite Differences. By Mr. George Harvey, of Plymouth.

1

I. SINCE A is the symbol which denotes the process of differentiation, let A1 be the symbol of the converse operation, by which the integral is obtained. Now, if u be any function whatever,

since

therefore A1. A u = A1 ▲ (u) = u, the primitive function.

Again, since A (u2) = 2 u ▲ u + Au2,

and that 2 u Au + Au2 is a function of u,

let it be denoted by O u, and therefore A (u) = Ou;

hence A' O u = A~' A (u2) = u2, the primitive function.

To present, however, a more general and comprehensive view of the subject, let u denote, as before, any function whatever; then

since

A-" denote the A"; and therefore

Hence, as ▲1 has been adopted to represent the converse operation of A, so let A-2, A-3, converse operations of A2, As as A, A2, A9

A" indicate the first, second, third,

and nth differences of the primitive function u, so will A1, A2,

A-" represent the first, second, third,

and nth integrals of the same function.

According to these principles, the formulæ (A) will become

Geometers, however, have adopted the symbols, 2, 2, Σ" as characteristics respectively equivalent to A-", and hence the equations

A-1, A-2, A-3,
Δ
denoted by (B) will be transformed into

COROLLARY.-Hence it appears that the nth integral of the nth differential of any function is the primitive function from which the

differential was derived.

II. It is moreover evident, from the preceding principles, that since

A may be considered as composed of the two factors

Δ

A, A, and is termed the second difference,
A may be considered as composed of the three factors
A, A, A, and is termed the third difference,

A may be considered as composed of the n factors ▲, A, A, ▲..... and is termed the nth difference

of the primitive function.

A-2 may be regarded as composed of the two factors

A-1, A-1, and is termed the second integral,

A may be regarded as composed of the three factors ▲-1, 4-, 4-, and is termed the third integral,

A

1

may be regarded as composed of the n factors
Δ', Δ', Δ', Δ' ... and is termed the
nth integral,

.....

or by adopting the characteristic E, as before,

Σ'

may be considered as composed of the two factors
Σ, E, and is termed the second integral,
may be considered as composed of the three factors
Σ, Z, Σ, and is termed the third integral,

Σ" may be considered as composed of the n factors
Σ, Σ, Σ, Σ..... and is termed the 7th integral,

of the primitive function;

of the primitive function.

COROLLARY.-Hence it appears that any differential, by the process of integration, may be changed into different forms; and also an integral may undergo corresponding variations by the process of differentiation: thus u representing any function, as before, its second integral may be denoted by either of the forms Σε κ = Δ u = AIA-1 u = A-3 A1 u Ξ Δ 5 A3 u = &c.; and its third integral by either of the forms

2

" Am u = A-" A" u = Am-n u = &c.

These combinations, it is obvious, may be varied without end. COROLLARY.-If in the latter form m be greater than n, the result will be a differential of the function u; but if m be less than n, then an integral of u will be obtained. Suppose, for example, m = 4, and n = 2,

then Σ"A" u = Σ2 A1 u = A2 u, the second difference of the primitive function u;

but if m = 3, and n = 5,

then Σ" ATM u = Σ5 ▲3 u = Σ2 u, the second integral of the primitive function u.

IV. A facility in the management and transformation of these symbols will be of great advantage to the student; and the following examples are therefore added to exercise his ingenuity :

-3

The second difference of E3 A" u = A2 Σ3 A" u = ▲2 A¬3 ▲" u =

The nth difference of Σ Δ" κ = Δ" Σ Δ" u = Δ" Δι Δ" κ =

2-2

The first integral of Σ A" u = A-1 A-1 A" u = ▲"~2 u = Σ"A" u = &c.

-2

The second integral of Σ A" u = ▲¬3 ▲ ̃ 1 ▲" u = A”—3 u =

The third integral of Σ A" u = A¬3 A−1 ▲" u = A*~4 u = Σ4 A" u = &c.

1

The nth integral of Σ ▲" u = `A” A1 A” u = ▲~1 u =

Σκ = &c.

The 7th integral of "A" u = A¬" A¬" A" u = A~" u =

Σ" u = &c.

The nth integral of "A" u = 4" A
Σm u = &c. *

— m An u = A

u =

It thus appears that the symbols usually employed in the differentiation and integration of functions are subject to the ordinary laws. of algebraic quantities; an idea which was first elicited by the genius of Arbogast.

By changing into d, and into, the preceding remarks are equally applicable to the theory of differentials.

*The utility and importance of these transformations must be obvious; for we can thus pursue the changes which the exponents undergo, through the several states of positive, zero, and negative; the positive state denoting the difference, zero a constant quantity, and the negative state the integral.