(p. 502) the area generated equals (T)-(Q), where (T) is the area of the given curve and (Q) the area of the closed Q-curve, or equal to twice the sector AOB, i.e., equal to l.a, where a denotes the arc OB. If we now could vary the proceeding so that (Q)=0 we should have (T)=la, and need only measure the arc a to get the area of the given curve.

To do this draw from A any straight line AX, place the planimeter in its original position OA, and move T from A along AX. The point Q now describes a tractrix OF with AX as asymptote. On doing the same from the end-position BA we get a second tractrix, BH, also with AX as asymptote.

If we now take on AX a suitable point R and draw with this point as centre and TQ as radius an arc ED between these two tractrices, we can consider the curve DOSBED as the Q-curve. We need only start at the position DR for QT, move T to A, then round the curve and back to R, turning QT at last about R till Q comes back to D.

If R has been well selected the arc ED will cut the curve OSB, and therefore the area of the Q-curve will be partly positive, partly negative. If, in fig. 16, the point R is moved higher up, the positive part will increase; on moving it lower down, the negative. Hence there must exist a definite position for R such that the area (Q) vanishes.

This shows the possibility of using the simple 'Hatchet' as a planimeter; but it is not yet a practical instrument. The above shows only:

If a point A, a line AX, and an initial position OA are taken arbitrarily, then a point R exists on AX such that on starting with T at R and Q at D we get the area reduced to twice the area of a sector with radius QT.

We have here at our disposal first the point A on the curve; secondly, the direction of the line AX; thirdly, the initial direction of QT, for it comes to the same whether we take OA or DR as given. These three

being fixed, the above proves the existence of a point R. The change of one of these three quantities alters the position of R.

We must, in order to get a practically useful rule for determining R, restrict the superabundance of choice which the above theory leaves us. A perfectly satisfactory rule has not yet been found. The only generally usable one is that given by the inventor.

As the Hatchet Planimeter has during the last few years excited some interest both in England and abroad, and as I have heard its invention attributed to various men, the following historical facts may be mentioned.

I became acquainted with it early in 1893 through Professor Greenhill, and ordered one from the maker, Herr Cornelius Knudsen, in Copenhagen. With it I received a pamphlet in French dated 1887. It contains an analytical theory without mentioning Captain Prytz. After my showing the instrument at the Physical Society and mentioning that a complete theory did not yet exist, there appeared in Engineering' a paper by Macfarlane Gray. In consequence of this Captain Prytz contributed his investigations to the same journal (where his name is changed to Pryty). It is practically an English translation of the pamphlet mentioned. I have since seen a new pamphlet with the name of Captain Prytz on it. In England the instrument seems, however, to have first become known through Mr. Druitt Halpin, who mentioned it to Professor Unwin, Professor Goodman, and other engineers about 1889. Professor Goodman

exhibited a somewhat improved form early this year at the Institution of Civil Engineers.

He has given the rod between the knife-edge and the tracer the shape of a circular arc, radius 7, and engraved a scale on it, so that it is possible to measure the arc between the two marks made on the paper instead of the chord.

Quite recently (October) I have learnt from Coradi that in December 1893 F. Hohmann communicated the idea of such a planimeter to him, and also that Professor Ljubomir Kleritj, of Belgrad, had invented a new planimeter. This is only a modification of Prytz's. In it the knife-edge is replaced by a knife-edge wheel, whilst the other end rests on two feet between which the tracer is so placed that its point is just off the paper. These feet are fastened to a cross-bar which is movable about a point above the tracer. The instrument thus rests firmly on three points. Both Professor Unwin and Coradi have pointed out to me that it would be an improvement if to Prytz's form a disc be added as a handle which can turn freely about the tracer. Kleritj's form supplies this.

Just now I have received a reprint from a Servian journal in which Professor Szily, of Budapest, has, at the request of Kleritj, developed the equation to the path of the knife-edge when the tracer moves along the circumference of a circle. It is dated December 1893. But it does not seem to advance the curiously interesting theory of the instrument.

The geometrical considerations given above were started by me and more fully worked out by Mr. A. Sharp, who has obtained several other results, which, however, do not yet yield any better practical rules than those given by Prytz.

LINKAGE INTEGRATORS.

J. Amsler (Vierteljahrsschrift,' 1856, p. 29) describes what he calls a 'Flächenreductor' (area-reducer), and in connection with this he gives a theorem about pantographs which deserves notice.

Starting with the fundamental theorem about the generation of an

area by a line of finite length, these theorems are easily obtained.

If we denote by (A B) the area swept over by the line AB during a cyclical motion, by (P) the areas enclosed by the close path of a point P, then we have

(AB)=(B)—(A)+nπAB2

If n=0, and this case alone we shall follow up, we have

(AB)=(B)-(A)

If we take a third point C on, or off, the line AB, we always have

(AB)=(B)–(A), (BC)=(C)—(B), (CA)=(A)—(C)

These give at once

(AB)+(BC)+(CA)=0 and (AB)=(AC)+(CB)

where account always must be taken of the sense, viz., it is

AB-BA and (AB)= (BA)

We also have, if a recording wheel be fixed on the line AB whose roll for a complete circuit is w,

(AB)=AB. w, (BC)= BC. w, &c.

Hence, if a line performs a cyclical motion without turning completely round, then the area swept over by any segment on the line is proportional to the length of the segment.

Let now A, B, P be points in a straight line and AB=a, AP =p, ..PB=a—p; then is

(AB)=aw=(B)—(A), (AP)=pw=(P)—(A)

On eliminating w we get

p (B)—a (P)=(p—a) (A), or (a−p) (A)=a (P)—p (B)

This is Holditch's Theorem.

From this it follows that we can always find one point P in the line AB which describes a curve of zero-area. For it

This point shall be called the zero-point in the line.

If (A)=(B), then is po, and the zero-point is at infinity.
For any other point P (not at infinity) we have now

a (P) a (A), .. (P)=(A)

If two points in a line describe closed curves of equal area, all points in the line do the same.

The area enclosed by any point in the line encloses always an area which is proportional to its distance from the zero-point.

In Amsler's planimeter, or any planimeter of Type I., the point Q is the zero-point of the rod' QT.

Let on this rod a point T' be taken so that QT=kQT'; then, whilst T' is guided along the boundary of an area (T), the point T will describe a closed curve of k times the area (T').

J. Amsler proposed (l.c., p. 29, 1856) to have a tube inserted at T' perpendicular to the paper. At the bottom this carries near the paper a glass plate with a small circular mark, and at the top a lens. The point T is now moved so that the mark at T' follows the small curve. The point T describes a closed curve whose area is registered by the wheel in square inches, say. The area (T') is therefore registered k times to the same scale. We have thus a planimeter which registers a magnified area, and is suitable to measure small areas. The advantage is this. In guiding a tracer round a curve the motion of the tracer will be more or less jerky. These jerks at T will be reduced at T'.

This arrangement, however, has one drawback. The figures described by T and T' are not similar, and this makes it difficult to guide T so that T' follows a given curve. But this can be overcome as follows: Let us consider a linkage as in fig. 17. OQT is an ordinary planimeter with

the pole at O. TQ is produced to D, and the extra bars OC, CD are added, with hinged joints. If T describes a closed curve, D will do the same, whilst Q and C are zero-points, provided that QT does not complete a

revolution. Let there be registering wheels on TQ and CD, and let their 'rolls 'during a circuit described by T be w and w'; then is

Hence the area (T) can also be measured by the roll of W'. Similarly, if DC be produced to T', the area of T' will be measured either by W' or by W.

If OQDC be made a parallelogram and T, O, T' are taken in a line, then T and T' will describe similar figures.

On this principle the firm Amsler-Laffon has recently constructed a

planimeter for measuring small areas.

For convenience of construction the points T and T' are made to describe figures which are only approximately similar, but sufficiently to make it easy to guide the tracer T so that T describes a given small curve.

It seems to me that we have here a new principle for making integrators which might be called linkage integrators; viz., if we take a linkage and move one point T along a given curve, then any other point T' will describe another closed curve whose area is dependent on that of the given

curve.

As a simple example, take a Peaucillier cell with fixed pole O (fig. 18). Give QT a wheel W and QT' a wheel W'. We have now two planimeters, OQT and OQT'. If OT=r, OT'=r', then we have always rr'=k2, where k2=OQ2—QT2.

If T describes a closed curve, T' describes another, and

(T)=aw, (T')=aw', where QT=a

2de if 0 is the angle which OT makes with a fixed line, and

Also (T)=¿fr2de

Hence, as T describes a closed curve, T' describes another whose area is proportional to

If ds denotes an element of the area (T), the last integral becomes by Stokes' Theorem about the conversion of a line- into a surface-integral

The origin must be outside the area (T) to avoid r=0.

On Methods that have been adopted for Measuring Pressures in the Bores of Guns. By Captain Sir A. NOBLE, K.C.B., F.R.S., M.Inst.C.E.

[Ordered by the General Committee to be printed in extenso.]

THE importance of ascertaining, with some approach to accuracy, the pressures which are developed at various points along the bores of guns by gunpowder or other propelling agent is so great that a variety of means have been proposed for their determination, and I purpose, in this paper, to give a very brief account of some of these means, pointing out at the same time certain difficulties which have been experienced in their employment, and the errors to which these methods have been in many cases subject.

The earliest attempt, by direct experiment, to ascertain pressures developed by fired gunpowder was that made by Count Rumford in his endeavour to determine the pressures due to different densities of charge. He assumed, the principles of thermodynamics being then unknown, that charges fired in a small closed gun-barrel would give pressures identical with those given by charges doing work both on the projectile and on the