As, however, with the lap-joint there is a double thickness of metal transversely, that joint is the strongest and stiffest portion to resist the stresses tending to bulge out the cylinder in the middle, and also to tear it into two halves. This affords some justification for the belief of old boiler makers, before rivetted joints were tried under a direct tensional load, that the joints are the strongest part of the boiler. And, indeed, this is what we find in practice. The thinest portion of the longitudinal furrows is generally exactly in the middle of the plate, and this is caused by the longitudinal stress, which is acting at right angles to the transverse cross bending stress. A strip cut from joint to joint is, in one respect, in the condition of a beam supported at both ends, uniformly loaded throughout its length, and, according to known principles, therefore giving way in the middle. (See Fig. 3.)

FIG. 3.

The middle ring of the boiler which burst on the Metropolitan Railway last year, and the fragments of which were examined by the writer, also first given way at a furrow. Captain Tyler reports that at from 16 to 19 inches from the transverse joint, or just about the middle of the plate, there was "very little metal left holding," while it gradually got to its original thickness of, as the groove receded from the centre of the plate and towards the transverse joints at each side. It is impossible to deny the existence of an infinite number of stresses acting on the sides of a vessel undergoing fluid pressure, producing what, for want of a better term, might be called a "bulging strain." Instances of this action may be noticed in the sketch of the leaden pipes given by Mr. Fairbairn,* which were bulged out in the middle by internal pressure, as also in the fire-box sidest influenced by the same means, and in the centre of the surface. Unaccountably enough, the effect of such a strain on the ultimate resistance, and, above all, the elasticity of materials, has been entirely neglected by investigators, and there are no published data on the matter. The effect of the internal pressure is evidently resisted by a double thickness of plate at the joints, so that the load on the middle of a single ring may be considered as determining the weakest part of the boiler. One of the rings of the Great Northern boiler which exploded on the Metropolitan Railway last May had a length transversely of about (say) 36 inches from lap to lap, with an inside diameter of 45 inches. If we now suppose a strip one inch broad cut from the 36-inch long plate, parallel to the longitudinal axis of the boiler, this strip is, supposing there be a pressure of 100 lbs. to the square inch, uniformly loaded with 3600 lbs., equal to a transverse load of 1800 lbs. at the

* "Philosophical Transactions, 1858," page 402.

(From Captain Tyler's report, dated 30th June, 1864, on the boiler explosion at the Overton Western Railway. The plate torn off is shaded, the course of fracture on the other side of furrow is shown by the thick horizontal line.)

station of the London & North

the boiler is dotted, while the

"Useful Information for Engineers, 1856." Appendix, xviii.

centre. Supposing the plate to form a true circle, a hoop one inch wide of the plate would be subjected, circumferentially, to a tensile load of 6000 lbs. per square inch, while (leaving out the diminution of the area at the ends through the flue tubes) each portion of the circle about 1 inch broad and 3-inch thick, is subjected to a load of about 1125 lbs. acting parallel to the axis of the boiler.

To construct a general rule or formula that would take into account the distorting effects of the lap or of the welt of butt-joints would be impracticable; but it is clear that the usual mode of calculating the strength of a cylindrical boiler from the tensile strength of joints tested by weights, or hydraulic pressure, directly applied, is far from being correct. It is only tolerably correct with scarf welded joints, or with butt-joints with outside welts. Even here, the hoop tension of the true cylinder is resolved into a cross bending strain, if the cylinder does not form a correct circle internally. The usual formula would be practically correct, if the boiler were prevented from altering its shape during the impulses sometimes given by the steam, and the quieter buckling action caused by the alternate increase and fall of the pressure. In fact, a boiler, like a girder, does not merely demand a high ultimate strength, but also a stiffness which is the protection against alternative strains-against buckling or collapse.

Disregarding the effects of the thickness of the material, a perfect cylinder should theoretically afford the same ultimate resistance, whether exposed to external or internal pressure. Its resistance to collapse should indeed be greater, as most materials give more resistance to compression than to tension. This is not the case, as the distortion of form progressively weakens an internal flue, by increas ing the load on its surface, while the contrary is rather the case with the boiler exposed to internal tension. Before Mr. Fairbairn showed the inherent weakness of flue tubes, their frequent explosions through collapse were ascribed to spheroidal ebullition and other similar causes. They are now, according to the engineer of the Manchester Boiler Association, stronger than the shells, by means of the T-iron and angle iron bands now generally used, and also by the excellent seams introduced by Mr. Adamson so long ago as 1852.* While T-iron and other bands could be used for the barrels of boilers not exposed to the fire, (as is recommended in Francet and by the Board of Trade Inspector of Railways,) Adamson's seams reversed would probably form excellent transverse joints for a shell fired from the outside, and, with a boiler like this, thin and narrow plates could be used, affording a stronger and tighter lap-joint. With a construction of this kind little or no deflection or bulging could occur, and the sectional area of the plate and the rings would really give the strength of the boiler.

* Specification No. 14,259.

Bulletin de la Société Industrielle de Mulhouse, 1861, page 532.

(To be continued.)

21

MECHANICS, PHYSICS, AND CHEMISTRY.

For the Journal of the Franklin Institute.

The Problem of the Gyroscope. By JAMES CLARK.

NEW YORK, June 24, 1865. Dear Sir: I enclose you herewith a solution of the problem of the gyroscope, which had occurred to myself before seeing any other, and which, if not novel in principle, I think will at least be found new in method. Further, I think it explains many things left obscure in most of the solutions hitherto published.

I am induced to offer it for publication at present, among other reasons, because the solution published in the New American Cyclopedia, towards the end of the article "Gyroscope," is palpably erroneous. The writer, in the first place, not recognizing the force generated by the rotation of the spindle about the pivot, as having anything to do with the supporting of the wheel; and, in the next place, he accounts for the rotation of the spindle by the fact that the revolution of the wheel is, on one side, against, and on the other, in conjunction with, gravity, both of which conclusions are demonstrably wrong.

Very respectfully,

JAMES CLARK.

The gyroscope, in its simplest form, consists of a wheel or disc revolving on an axle, which rests on an upright pivot, as represented in the annexed figure. When the wheel is set in rapid revolution, and the end of the axle or spindle opposite that on which the wheel revolves is placed on the pivot, the wheel and the end of the axle on which it revolves, will remain supported as indicated in the diagram, and the axle or spindle will rotate about the pivot in a direction opposite to that in which the top of the wheel revolves on the axle.

An analysis of the forces producing these phenomena, is what is required by the problem of the gyroscope.

A

IN

MI

First. What forces produce the revolution of the spindle Ao about

the pivot A?

If we begin with the particle of matter situated at the point c in the circumference of the wheel, we will find it acted upon by two forces -the centrifugal, which tends to impel it in the direction of the tangent c D, and that of gravity, modified by the suspension of the axle on the pivot AB, the tendency of which is to cause the particle to describe the arc CH. The resultant of these forces, by a well known mechanical law, gives to the particle at c an impulsion in the direction of the line c E, diagonal to the tangent C D and the arc CH*; and the like may be predicated of every particle of the wheel above the horizontal diameter T L. The resulting tendency of the forces acting on all these particles, then, is to change the plane of revolution of the wheel to one diagonal to the two planes perpendicular to the horizon, which respectively contain the tangent CD and the arc C H. Now, the wheel can only assume this new plane of revolution, or one parallel to it, by moving backward in a direction opposite to that in which the top of the wheel revolves, which it is free to do by the rotation of the spindle about the pivot a. If we consider the particle at Y, (and the like is true of every particle situated below the diameter T L,) we find it likewise acted upon by the centrifugal force in the direction of the tangent YR, and by gravity, (modified as explained above,) in the direction of the arc Y P. The result is a tendency to produce motion in the direction of yo, the diagonal between the direction of these forces, and, consequently, (YQ being parallel to CE,) to change the wheel's plane of revolution into the same plane into which it is impelled by the forces acting on the particles situated above the horizontal diameter.

We have now accounted for the rotation of the spindle about the pivot a B.

Secondly. It remains to account for the paradox of the wheel's continuing supported in the air.

If we take the particle situated at T, (and the like is true of every particle situated on the same side of the perpendicular diameter c Y,) we discover it is acted upon by the centrifugal force, in the direction of the tangent T w, and by the force communicated by the rotation of the spindle about the pivot A B, in the direction of the arc T S. The result is a tendency to impel the particle at T in a line TV, diagonal to the tangent and the arc, and the wheel into a plane of revolution diagonal to the horizontal plane which contains the arc TS, and the plane perpendicular to the horizon and to the axle A 0, which contains the tangent T w. But the wheel can only assume this new plane of revolution, or one parallel to it, by the elevation of the extremity o of the spindle on which it revolves. In like manner, the particle situated at L, (and the like is true of all particles on the same side of cY,) is It is, perhaps, not strictly correct to speak of a line as diagonal to a right line and a curve. It would be more exact to say, that the resulting impulsion given to the particle at C, is in the direction of the diagonal between the tangent C D and a tangent to the arc c H at c.

†The spindle is here assumed to be placed on the pivot at right angles with it, which is not necessarily the case; and the angle at which it is placed may subsequently vary, accordingly as the elevating tendency is greater or less than the force of gravity.

acted upon by forces tending severally to impel it in the direction of the tangent LN, and of the arc LK. The evident result is a tendency to impel it in the direction L M, and to produce the same change in the plane of revolution of the wheel, that we have already shown to be the effect of the forces acting on the particles on the other side of cy; and it is this constant effort to change the plane of revolution into one which, or a plane parallel to which, the wheel can only assume by elevating the extremity of the axle, which prevents it from falling.' The whole may be thus summed up.

*

The rotation of the spindle on the pivot, is the resultant of the action of gravity and the centrifugal force generated by the revolution of the wheel on the axle; while the wheel is prevented from falling by the combined action of the same centrifugal force and the rotation of the axle on the pivot.

COROLLARY 1. If the extremity of the spindle opposite the wheel, be prolonged beyond the pivot in the direction A X, and a weight be placed at X somewhat more than sufficient to balance the wheel, the spindle, instead of rotating about the pivot in a direction opposite to that in which the top of the wheel revolves, will rotate in the same direction. COR. 2. If the weight exactly balance the wheel, the spindle will cease to rotate about the pivot.

COR. 3. If the weight be removed, and the rotation of the spindle be stopped by the interposition of any obstacle or opposing force, the wheel will fall.

COR. 4. If the weight be suffered to remain, and be more than sufficient to balance the wheel, and the rotation of the spindle be stopped, the wheel will tilt upward.

OBSERVATION. Another fact remains to be considered, viz: the resistance offered by a body in motion to any force tending to divert or deflect it from the direction in which it is moving.

Suppose a body placed at A, to be impelled in the direction AB, by a force sufficient to carry it to B in a given time. Now, if the same body be simultaneously acted upon by another force in the direction A D, sufficient to carry it to

D in the same time, by a familiar law already alluded to, the body will move in the diagonal A C, and will reach c in the same time in which the forces acting singly

would carry it to B or D. The point to which attention is invited is this, that while the second force, if acting alone, would impel the body in the direction A D, when the same force acts in the same direction on the body simultaneously impelled toward B, with a force proportioned to that acting in the direction of D, as A B to A D, the path A C described by the body, more nearly coincides with the direction of the *If the spindle be placed on the pivot so as to form either an acute or obtuse angle with it, it is evident that the centrifugal force generated by the rotation of the spindle about the pivot, will tend to bring the spindle at right angles with the pivot; in the former case, acting in aid of the elevating force, in the latter, in opposition to it: