The object of the present paper is to show how, by graphic methods, we may find the relation between the effort exerted at one part of a machine in motion, and the resistance overcome at another part; the solution found is rigorous for all motions in one plane, and takes count of the friction, weight, and inertia of the parts. It also takes into account the stiffness of ropes or belts.

The paper shows that we may represent any machine, at any given instant, by a frame of links, the stresses in which are identical with the pressures at the joints of the machine. This self-strained frame is called the dynamic frame of the machine, and may be so drawn as to represent the machine either rigorously, taking into account friction, weight, inertia, and rigidity, or approximately, omitting some of the conditions under which the machine works.

Moreover, it is shown that for all machines (in which the motions can be represented as in one plane), the dynamic frame is of one type, either simple or compounded. The dynamic analysis of machinery into parts represented by this simple frame is believed by the author to be novel. It is consistent with the kinematic analysis of Reuleaux.

Page 190, to the alinea (beginning with): “the sole condition to be satisfied, is,” &c. (ending with): “… the versor of the expression may then become what it may,” add:

provided this versor does not differ from the versor of the product

made by successive multiplication (according to the definition of multiplication of more than two vectors), the quaternions w + ω, w−ω being likened respectively to the products

by a proper choice of the vectors p and .

In a paper on “The Bifilar Magnetometer, its Errors and Corrections” &c., which appeared in the Transactions of the Royal Society of Edinburgh, vol. xxii. p. 467; it was shown that, in turning the apparatus which carries the upper extremities of the two wires through an angle exceeding by ν that through which the lower extremities and the magnet are moved, the upper end of each wire is also turned through an angle greater by ν than the lower end. The whole force acting on the magnet, to turn it from the meridian, is compounded of the bifilar and the unifilar torsion—only the former of which had been considered in the method issued by the Committee on Physics of the Royal Society of London for the instruction of directors of observatories.

It was sought in the paper cited to show the change that the neglected forces would introduce into the expression for the unit coefficient, which was found to be

instead of

where Δν is the arc value ot one scale division of the instrument, and β is the variation of ν due to the unifilar torsion. There were errors of copying, including omissions in the investigation given. I also remarked, after the paper was printed, that the value of k found was less than when the unifilar torsions were neglected (since cot (ν + β) < cot ν); whereas all the determinations by other methods, in which the unifilar torsions were included, had shown a greater coefficient.

Certain cinematical and physical questions lead to the problem: to determine the directions in which a given linear vector-function, ϕρ, assumes a direction parallel to that of the vector, ρ, on which it depends.

The condition of parallelism is expressed by

and it is translated into the equation

where g represents a certain scalar, on whose determination the whole problem depends.

Following the method traced out by Hamilton, we treat this equation successively by

α,β,γ, being any system of vectors not coplanar between each other; but we will state at once that throughout the whole of this paper we shall assume α,β,γ, to form a system of treble rectangular unit-vectors, of which hypothesis the justification is evident.

Towards the end of last winter session, a Memoir by me, on the “Parallel Roads of Lochaber,” was read to this Society, and it has since been published in our Transactions. The subject was far from being exhausted. Nevertheless, I had no intention of continuing the inquiry, venturing to think, that enough had been adduced by me to support the conclusions at which I had arrived. But during the course of last summer, Dr Tyndall of London visited Lochaber. He went for the special purpose of studying the “Roads,” and of enabling him to give a public lecture regarding them in the Royal Institution, Albemarle Street, on 9th June.

In the course of his lecture, Dr Tyndall alluded to my recent Memoir, and also to researches described in a previous Memoir. But he dissented from my solution of the problem, and told his hearers (I quote his words) that they might “with safety dismiss it (the detrital barrier theory) as incompetent to account for the phenomena. The theory which ascribes the Parallel Roads to lakes clammed by barriers of ice, has, in my opinion, an amount of probability on its side, which amounts to a practical demonstration of its truth.”

These views having been rested on observations made in the district by Dr Tyndall himself, I felt that it was only due to a person of his scientific reputation, to reconsider my own opinions, and to weigh well his reasons for coming to a different conclusion.

(I.) Let F(x) = 10 + b1x + … + bnxn = 0 be an equation whose roots are all real and different; then

where ϕ(x) = 1 + a1x + … + amxm and Σ indicates summation for all the roots of F(x) = 0: of which r is a root.

Let also

then

and equating coefficients of like power of x,

whence

the suffix number indicating the order of the determinant.

In Crelle's Journal for 1844, at the end of a paper on Cubic Forms, Eisenstein gives the following results:—

where m is any odd number and ρ a primitive root of the equation zm = 1;

where

No demonstrations are given of these identities; but they are said by the author to be only particular cases of a very general equation, the complete theory connected with which he hoped to give on another occasion.

At p. 193 of the same volume of Crelle he returns to the subject, not however for the purpose of giving the promised theory, but to add several other results similar to those before given.

The following paper contains, in a compact form, the substance of several somewhat bulky communications laid before the Society during the present session. The gist of each of these separate papers will be easily seen from the abstracts given in the Proceedings. These contain, in fact, many things which I have not reproduced in this digest. Nothing of any importance has been added since the papers were read, but the contents have been very much simplified by the adoption of a different order of arrangement; and long passages of the earlier papers have been displaced in favour of short general statements from the later ones. With the exception of the portion which deals with the main question raised, this paper is fragmentary in the extreme. Want of leisure or press of other work may justly be pleaded as one cause; but there is more than that. The subject is a very much more difficult and intricate one than at first sight one is inclined to think, and I feel that I have not succeeded in catching the key-note. When that is found, the various results here given will no doubt appear in their real connection with one another, perhaps even as immediate consequences of a thoroughly adequate conception of the question.

The construction of the toothed wheels used in machinery gives rise to some very interesting investigations in the geometry of motion. The general problem is so to shape the contours as that they shall remain in contact while the wheels turn on their centres with uniform angular velocities.

The inquiry becomes more extensive when the velocities of the wheels are to be variable; as, for example, when we seek to imitate the revolutions of the planets round the sun, and for that purpose introduce the equation of the centre.

In these cases the wheels are supposed to turn on fixed centres; but we may still farther extend the scope of our researches by removing the centres and subjecting the discs to the single condition that they roll upon each other.

If two discs A and B touch at the point S, and if they so move as that the point of contact shifts equally along the two boundaries, they are said to roll on each other; that is to say, if we measure equal distances ST, SV along the two boundaries, the points T and V will come together in the course of the movements.

Before submitting the results of my examination of the silicates, it is incumbent upon me to give an outline of the processes adopted for their analysis, seeing that the amount of confidence which is to be placed in the results must rest very immediately thereupon.

1. Determination of the Water. — In all silicates which were found, after thorough drying, to give out moisture when heated in the closed tube, the amount of loss upon ignition was calculated as water. About 2 grammes, crushed in the diamond mortar, were first merely heated throughout in the air-chamber of a water-bath, the contact temperature of which was 212° Fahr., the temperature of the air-space being never below 199°. The weight after this heating was taken; and, except in the case of a mineral functioning in any peculiar way, the loss in weight was estimated as hygroscopic moisture. The mineral was now continuously heated in the bath until the weight was found to be constant. It was then exposed for the space of one hour to a full red heat, in the flame of a Bunsen burner, after which it was re-weighed. Finally, it was exposed for ten minutes to a heat approaching whiteness, in a three-jet Griffin blast-furnace, again re-weighed, and again heated in this furnace, until the weight was constant.

If light, emanating from a fixed source, be reflected to the eye, also fixed, from the surface of a polished cylinder, which cylinder changes its position in some definite manner, the point of reflection moves in some curve or locus whose nature may be made the subject of investigation.

In the present paper I propose to examine that case in which an indefinitely thin cylinder is restricted to pass through a fixed point. The locus of the point of reflection is then a curved surface, For the present I shall still farther restrict the polished line to the plane passing through the vertex, the source of light, and the eye ; the locus being then a plane curve.

Let then a fine polished straight line, extended indefinitely both ways, turn on the vertex O, while light emanating from the source A is reflected to the eye at B; it is required to investigate the locus of the point C at which the reflection takes place.

The reflection is only possible while the polished line passes outside of the angle AOB. When the wire lies along AO, the point of reflection is at A ; when along OB, at B; and when it is equally inclined, outwards, to OA and OB, the reflection occurs at O, wherefore the curve must pass through the three points O, A, and B.