Another Method of finding the Action of the Triangle vrm.

The expressions we have found will terminate only when m is one of the series of numbers 2, 4, 6, &c. If m is among the odd numbers 1, 3, 5, &c. + will be a whole positive

2

number; and we have for the action of the triangle, or rectangle (accordingly as the fluent, with respect to t, is taken from to, to trT, or from to, to ty') provided m is greater than 3,

I

A = faT {(m—1) (4a2+T2)

X

t

(a2+ T2+t2)?

t

(m-2) (m—4)

1

(e2+7°•+8)=3 + (m=1) (m−3) (m=3)}{@+T) × (e+7++)=

(a2+T2+t2)=

(m—5) (a2+T2)3

t

+

+

(m−2) (m—4) (m—1) (m—3) (m—5) (m—2) (m—4) ... (m—1) (m—3) (m—5) ........ 2 (a2 + T2)TM

*****..

3

X m

}

a2+1

Let us, for brevity, denote this quantity by saϢ (a, m, T, t); then for the action of the triangle vrm we have A = faϢ (a, m, T, rT); and for the rectangle, whose sides are rm (y) and rv (y') A = saϢ (a,m,T,y'). When m=1, or 3, the above expression will not give the attraction; but we evidently have, in the case of m=1,

..........

A =√x arc (tang.

ar
a2+T2

+

2

m-2

(m—1) (m—3) (u2+T2)2

........ 2 (a2+T2)TM

a2+T2+ t2
a2+1

+

a2+ × arc (tang. =

+T

; and when m = 3,

Cor. 1. For a polygon of n sides, these expressions must be multiplied by 2n as usual; and when m is greater than 3, A = 2nfaχ (a, m, T, rT), if in this we make n infinitely great and r infinitely small, it ought to enter into the general case of the attraction of a circle given in Cor. 1, to the first part of the lemma: and in fact we get

m-2

+

(m-2) (m-4)
(m—1) (m—3) m−5)

2narTT

(m-2) (m-4)...... 3
(m—4)
(m-1) (m-3)...... 2

} } × S

X

A = { // — + M-I (m-1((m-3) (m-2) (m-4). (m−1) (m−3). (a2+T2)TM-|-1 cause the quantity between the brackets is plainly equal to unity, which is the same form as was becomes A =√2nraT'İ

2

(a2+T2)#‡i found before for the general case.

Cor. 2. When r becomes infinite, and the triangle rmv is changed into an infinitely extended rectangle, we have for its attraction

+

A

ani

· (a2 + T2)=='

except when m=1, in which case,
in which case, A =S

........ 2

3.5.7......... (m −2)
2.4.6...
(m-1)

+ .........

από ✔a+T2

+

, or, be

Scholium.

This lemma has been treated on the supposition that the density is the same at every part of the triangle rmv, fig. 1; but there are other hypotheses which render the solution easier: for instance, we may conceive the density of a particle at q to be as its distance (t) from the line rm, in which case aiti (a3+ T2+12)”‡1

A = SS;

=S{;

- at
(m—1) (a2 + T2 + 42)*—*

+

at

(m—1). (a2+T2)2

where t must be made rT or y' accordingly as the action of a triangle or rectangle is required.

From this simple case, we may not only arrive at some curious results, connected with the particular hypothesis of density, but may find with equal ease the figure of a homogeneous solid of revolution of greatest attraction, as will just now be seen.

If the density was to be a function of a and T only, it would be sufficient to multiply the values found in the lemma, by that function, see lemma 1.

Prop. 35.

Let Prop. 31 be again proposed, but with this difference, that the force is inversely as the mth power of the distance, and that the density of any particle of the cylinder is as its distance (t) from that middle section (parallel to the ends of the cylinder) which passes through the attracted point.

In the expression we just now found, in the preceding scholium, put x for a, and d (half the length of the cylinder) for t. The action of the cylinder is

2

-xxT xxi A = 4SS { + (m− 1) (x2+T2 + d2)TM— 1 (m−1)(x2+T2)? its quantity of matter is 4√ff*Ïtt = 2ffxİt; so that we have, for the equation of the curve bounding the base,

x

+ Cd'=0.

x

(x2 + y2)TM—— (x2+y2+d2)”—1

2

Cor. 1. When dis infinitely small, this becomes

M-I
2

2

xd2

(2*+3)=1+ Cd, or (++C′ = 0.

2

(x2+32)TM+:

Let a be the value of x when y = 0, then C' =

;

; and

the equation of the curve, bounding the plane of greatest attraction, is

a"x=

(x* + 3•)*‡',

which is exactly the same result as that obtained by Mr. PLAYFAIR, p. 203, on the supposition of homogeneity; and this was to be expected; for, though a certain condition of the density of the cylinder entered into the foregoing problem, yet when d vanishes, and the solid becomes a plane, we must evidently obtain the same result as if it had been arrived at by supposing the cylinder homogeneous; which in fact it will be when the length is evanescent.

Nor is this observation to be confined to that particular case when the density is as t: if we had solved the problem on the supposition of any function of x, T, and t, for the density, it is easy to see that though different functions will give different results when dis finite, yet when the solid becomes a plane, and do, the equation will always be reduced to

(x2 + y•)TM+1.
y2)

2

a"x=

Hence we may conclude, that, the solid of revolution which shall exercise the greatest attraction on a point in its axis, when the force is inversely as the mth power of the distance, and the density either uniform, or any function whatever of x and T (T being the perpendicular let fall from any particle to the axis of the solid, and x the distance between the foot of that perpendicular and the attracted point) will have, for the equation of its generating curve, (x* + y• ) *+*.

ax=

Cor. 2. Nothing can be learned from the equation

x

x

(2*+y)==" ̄ ̄ (P+y*+d')== + Cd*=o,

(x2+y2+

when m1. The curve is then transcendent, and has for its equation xL. (x2 + y2 + d2) — xL. (x°+yo) + Cď = 0. Cor. 3. If the cylinder becomes infinitely long, (m being positive and greater than unity) the equation of its base is + C'= 0;

x

{x® +y®)TM—1

let a be the value of x when y = 0; then C'== equation becomes

X

(.x2+y2)===

= 1,

x2 If m2, as in the case of nature, this becomes x2+ y2 so that the infinitely long cylinder of greatest attraction will be an infinitely long rectangle, with its edge turned to the attracted point.

If m = 3, we have ax = x2+y', the equation of a circle with the attracted point in its circumference.

If m = 4, the equation is ax = (x2 + y2), which is Mr. PLAYFAIR'S curve of equal attraction.

If we want the figure of the infinite cylinder of greatest attraction, when m=1, we must have recourse to the last corollary; where we found

I

, and the

xL (x°+y+d3) — xL (x2+ya) = C'. This, when d is infinite gives xL. d = C', or, x = const., the equation of a plane perpendicular to the axis of x.

Cor. 411 If we would solve Proposition 34, but with this difference, that the force is now inversely as the mth power of the distance, and the density, in the generating rectangle uvv'u', fig. 15, is, at any point, as its distance from rm or y; we need only put f (x) (given by the nature of the curve pr) for d, in the equation here found, and we get that of pr, in the case of