one of the four angles bx, cx, px, qx by two proportions which when

united give these four values of x, viz.

The Angular Calculus was applied with advantage to the resolution of Quadratic Equations, first, I believe, by Dr Halley, in Lectures given at Oxford in 1704. From this it might be inferred that it may be applied to the solution of every Algebraic problem which produces a Quadratic Equation, without a previous reduction to that form, although I do not know that this application has been expressly treated of, and examples given. The formulæ which have been investigated in this paper apply with peculiar advantage to the solution of a known problem in Algebra, which appears at first sight to be by no means easy. It is this:

Find x, y, and %, from these equations

x2 + xy + y2 = c2,

x2+xz +≈2 = b2,

y2 + y z + z2 = a2.

Here a, b, c are given numbers.

These equations become identical with the equations of Art. 10. or 18, if we suppose the angles a, ß, y to be all equal, and each 120o, because then 2 cos a= −1. 1. Therefore A, B, C being the angles of a triangle whose sides are a, b, c, and

Here a' may be any number, provided we take

With these logarithms, proceeding exactly as in the calculation Art. 23, we find x=7, and hence, as in Art. 21, y=9, x=10.

We might find x otherwise from formula VII. Art. 18, which may stand thus:

and it is a remarkable property of this expression that it is exactly similar to the equation

V. Mathematical Considerations on the Problem of the Rainbow, shewing it to belong to Physical Optics. By R. POTTER, ESQ. of Queens' College.

[Read Dec. 14, 1835.]

HAVING lately, in the course of my academical studies, had occasion to read more carefully the theory of the Rainbow, I was convinced of the inadequacy of the popular mode in which it is treated in elementary books (this is Sir Isaac Newton's explanation, see his Optics, p. 147). The reason which is given, why the various prismatic colours are seen, each, so brilliantly in the rainbow, is that the intensity of any colour fades away so rapidly from its maximum, that it does not prevent, in any great degree, the other colours being

seen as such.

This is clearly an arbitrary assumption, which will not bear examination, for within the primary bow, the grayish light is of very considerable intensity when the display is a fine one, and in a like proportion in fainter displays. So that we should naturally expect the brilliant colours yellow and green (even laying aside the red and orange) to have still an intensity at the places of the indigo and violet, sufficient to drown the effect of colour in those weak shades.

We find that this is the fact in certain cases, as in Fog bows and similar appearances, when the size of the aqueous spheres is very small, but in other cases the violet especially is very bright. The popular theory, however, offers no reason why the size of the spherical drops should influence in any degree the colours of the bows, and it

is only in referring the problem to Physical Optics, and considering the interference of the light which arises, that we understand how the size of the drops varying and determining the angular positions of the bright and dark fringes for any colour, causes the appearances to vary, by modifying the extent of the overlapping of the various colours.

The existence of the supernumerary bows furnishes a still more weighty objection to the common explanation; which supposes only one maximum of intensity for each colour, at the angle at which two consecutive rays emerge from the drop parallel to each other. But these supernumerary bows shew that the true explanation must furnish reasons for a succession of maxima and minima for each colour, and this the principle of interference does in a manner perfectly in accordance with recorded observations of the phenomena.

It was not until after I had finished the mathematical investigations, that I learned from Mr Whewell, that Dr Young had previously applied the principle of interferences to explain the supernumerary rainbows. If I had known this earlier, I probably should never have entered further into the subject; as it is, it will be found that I have shewn a method of obtaining the result, which I am not aware that he has any where given; and at any rate I shall have awakened the attention of mathematicians to this interesting phenomenon.

Dr Young's account is very concise, and insufficient as a mathematical explanation; he, however, notices the brilliancy of some of the colours being assisted by the interference of the others. At page 470 of his Lectures on Natural Philosophy, he says, “We have already seen that the light producing the ordinary rainbow is double, its intensity being only greatest at its termination where the common bow appears, while the whole light is extended much more widely. The two portions concerned in its production must divide this light into fringes; but unless almost all the drops of a shower happen to be of the same magnitude, the effects of these fringes must be confounded and destroyed; in general however they must at least co-operate more or less in producing one dark fringe, which must cut off the common rainbow much more abruptly than it would otherwise have been terminated, and