forms are similarly modified, so as to produce a series of what are called hemihedral forms.

Apparent but Congruent Exceptions.

Not unfrequently a law of nature will present results in certain circumstances which appear to be entirely in conflict with the law itself. Not only may the action of the law be much complicated and disguised, but it may in various ways be reversed or inverted, so that all careless observers are misled. Ancient philosophers generally believed that while some bodies were heavy by nature, others, such as flame, smoke, bubbles, clouds, &c., were essentially light, or possessed a tendency to move upwards. So acute and learned an inquirer as Aristotle entirely failed to perceive the true nature of buoyancy or apparent lightness, and the doctrine of intrinsic lightness, being expounded in his works, became the accepted view for many centuries. It is true that Lucretius was fully aware why flame tends to rise, holding that

The flame has weight, though highly rare,

Nor mounts but when compelled by heavier air.'

Archimedes also was so perfectly acquainted with the buoyancy of bodies immersed in water, that he could not fail to perceive the existence of a parallel effect in air. Yet throughout the early middle ages the light of true science, clear though feeble, could not contend with the powerful but confused glare of the false Peripatetic doctrine. The genius of Galileo and Newton was required to convince people of the simple truth that all matter is heavy, but that the gravity of one substance may be overborne by that of another, as one scale of a balance is carried up by the preponderating weight in the opposite scale. It is curious to find Newton gravely explaining

the difference of absolute and relative gravity, as if it were a new discovery proceeding from his theory. More than a century elapsed before other apparent exceptions to the Newtonian philosophy were explained away.

Newton himself allowed that the motion of the apsides of the moon's orbit appeared irreconcilable with the law of gravity, and it remained for Clairaut to remove the reproach by more complete mathematical analysis. There must always indeed remain, in the motions of the tides or of the heavenly bodies, discrepancies of some amount between theory and observation; but like discrepancies have so often yielded in past times to prolonged investigation that all physicists have come to regard them as merely apparent exceptions, which will afterwards be found to be new confirmations of the law with which they now seem to conflict.

The most beautiful instance, perhaps, which can be adduced of an apparent exception, is found in the total reflection of light, which occurs when a beam of light within a medium falls very obliquely upon the boundary separating it from a rarer medium. It is the general law that when a ray strikes the limit between two media of different refractive indices, part of the light is reflected and part is refracted, but when the obliquity of the ray within the denser medium passes beyond a certain point there is a sudden apparent breach of continuity, and the whole of the light is reflected. A very clear reason can be given for this exceptional conduct of the light; for according to the law of refraction the sine of the angle of incidence always bears a fixed ratio to the sine of the angle of refraction, so that the greater of the two angles, which is always that in the less dense medium, may increase up to a right angle, but when the media differ in refractive power, the less angle cannot become a right angle, as this • 'Principia,' bk. II. Prop. 20. Corollaries, 5 and 6.

e

would require the sine of an angle to be greater than the radius. It might seem, perhaps, that this was an exception of the kind elsewhere described as a limiting exception, in which a law is shown to be inapplicable beyond certain defined limits; but in the explanation of the exception according to the undulatory theory, we find that there is really no breach or exception to the general law. Whenever an undulation strikes any point in a bounding surface, spherical waves are produced and spread from the point. The refracted ray is the resultant of an infinite number of such spherical waves, and the bending of the ray at the common surface of two media depends upon the comparative velocities of propagation of the undulations in those media. But if a ray falls very obliquely upon the surface of a rarer medium, the waves arising from successive points of the surface may spread so rapidly as never to intersect, and no resultant wave will then be produced. We thus perceive that from general mathematical conditions may arise very distinct apparent effects.

There may occur from time to time distinct failures in our most well-grounded predictions. A comet, of which the orbit has been well determined, may fail, like Lexell's Comet, to appear at the appointed time and place in the heavens. In the present day we should not hold such an exception to our successful predictions to weigh against our belief in the theory of gravitation, but should assume that some unknown body had through the action of gravitation itself deflected the comet. As Clairaut remarked, in publishing his calculations concerning the expected reappearance of Halley's Comet, a body which passes into regions so remote, and which is hidden from our view during such long periods, might be exposed to the influence of forces totally unknown to us, such as the action of other comets, or even of some planet too far removed from the sun to be ever perceived by us. In the case of

Lexell's Comet it was afterwards shown, curiously enough, that its appearance was not one of a regular series of periodical returns within the sphere of our vision, but a single exceptional visit never to be repeated, and probably due to the perturbing powers of Jupiter. Yet this solitary visit was a strong confirmation of the law of gravity with which it seemed to be in conflict.

The division of Biela's Comet into two companion comets was at the time when it occurred one of those unlookedfor and inexplicable events which awaken the attention and interest of observers in the highest degree. Comets indeed have altogether the character of eccentric strangers intruding into our planetary system, and in almost every point they are yet inexplicable; but there is a possibility that the separation of Biela's Comet may prove to be a comparatively ordinary event of cometary history. For if, as is now believed, comets be aggregates of small meteoric stones or particles, forming the denser parts of continuous streams of such bodies circulating round the sun, then it is not unlikely that these aggregates may at times be increased or diminished by the meeting or separation of meteoric streams.

Singular Exceptions.

Among the most interesting of apparent exceptions are those which I propose to call singular exceptions, because they are more or less closely analogous to the singular cases, or solutions which occur in mathematical science. A general mathematical law embraces an infinite multitude of cases which have a perfect agreement with each other in a certain respect. It may nevertheless happen that a single case, while obeying the general law, stands out as apparently different from all the rest. The daily rotation of the earth upon its axis gives to all the stars

in the heavens an apparent relative motion of rotation from east to west; but out of countless thousands which obey the rule the Pole Star alone seems to break it. Exact observations indeed show that it also revolves in a small circle, but it might happen for a short time that a star existed so close to the pole that no appreciable change of place would be caused by the daily rotation. It would then constitute a perfect singular exception; for, while really obeying the law, it would break the terms in which it is usually stated. In the same way the poles of every revolving body are singular points.

Whenever the laws of nature are reduced to a mathematical form we may expect to meet with singular cases, and, as all the physical sciences will meet in the mathematical principles of mechanics, there is no part of nature where we may not probably encounter them. In mechanical science itself the circular motion of rotation may be considered a single exception to the rectilineal motion. of translation. It is a general law that any number of parallel forces, whether acting in the same or opposite directions, will have a resultant which may be substituted for them with like effect. This resultant will be equal to the algebraic sum of the forces, or the difference of those acting in one direction and the other; it will pass through a point which is determined by a simple formula, and which may be described as the mean point of all the points of application of the parallel forces (vol. i. p. 422). Thus we readily determine the resultant of parallel forces, except in one peculiar case, namely, when two forces are equal and opposite but not in the same straight line. Being equal and opposite the amount of the resultant is nothing, yet, as the forces are not in the same straight line, they do not balance and destroy each other. Examining the formula for the point of application of the resultant, we find that it gives an infinitely great magnitude,