27t 2cq cot a , p - q cota ol. – cost. (et – B + 5–1 –;= *): and the integral of this, taken with regard to a between the limits before mentioned, is Let us now refer the image on the screen to polar co-ordinates, which will be done by writing r cos 0, r sin 9 for p, q respectively. For brevity, b *:::, then the above expression for the disturbance at M may be written By expanding the numerators of these fractions, and arranging the result in two terms containing respectively sin V and cos V, this ex pression for the disturbance at M may be written in the following form : The intensity of the light at the point M, as is well known, is equal to the sum of the squares of the coefficients of sin P and cos P', ordinates r and 6. When the triangle is equilateral a = 60, and the equation for the brightness assumes the very symmetrical form The Interpretation of the Formula for the Brightness. IT will be found upon trial that the value of Z is not altered when –6 is written for +6; and hence it follows that the light is symmetrically arranged with regard to the axis of a. It will likewise be found that the value of Z is not affected when any one of these values, 6 + 60°, 9 + 120°, 6 + 180°, 6 + 240°, 6 + 300", is substituted for 6 ; and hence it follows that if from the origin of coordinates, or centre of the screen, six lines be drawn upon it making respectively the angles 0, 60", 120°, 180", 240', 300" with the axis of a, or, which is the same, inclined at angles of 60° to each other, the light upon the screen is similarly and symmetrically arranged with regard to every one of them. Wherefore, the light being symmetrically arranged about these six lines, it will only be necessary to examine our formula for Z between the values 6 = 0 and 6 = 30". It will at once be seen from an inspection of the equation preceding the one marked (B), that the value of Z depends upon the three terms each of which is precisely similar to the principal term in the expression for the intensity of light in the experiment of Fraunhofer's gratings; and at first sight it might be deemed sufficient to examine each of these terms separately, and thence judge of their united effect: but it will be found upon trial that the multipliers by which they are connected toge - ther exercise such an important influence upon their values, as to render this method utterly inapplicable in the present instance. Thus, if 6 be very small, the second and third terms are very nearly equal, and having different signs their sum is very small; but being afterwards divided by sin 9, the quotient is large; and their united effect is as great as that of the first term. From the necessity which thus exists of taking in at once the whole of the expression for Z at every step of our examination, we shall be obliged to feign several cases, and effect corresponding expansions and reductions for each; and from these particular results infer, in the best manner we are able, the general appearance and brightness of the image upon the screen. 1. Let us suppose r and therefore m extremely small. This will be true of parts very near the centre of the screen. In this case we must expand the expression for Z in a series of terms arranged according to the powers of m. This may be effected most readily as follows. And that part of the expression for Z which is enclosed within the brackets is equal to X (*!) By the usual method of finding the sums of the powers of the roots of equations, we easily find Wherefore, by substitution, we finally obtain 3a* m* m" (81 + 4 sin 36) m' Z=#1-# * gid -*.G.H. f. 1 ...} 3 a” Tr" | r \* a'T' (r) (81 + 4 sin’30)a'T'/r)" =#4 #1-# (") + so (o) –"...", (t)+...} A striking feature of this series is, that its leading terms are entirely independent of 6, and therefore while r is so small as to allow the series to be represented by its first three terms, the brightness will be independent of 9: and therefore consecutive circles of uniform brightness will surround the centre. When r is = 0, the brightness = 16 °. which is independent of X, and therefore the central point of the image is white. When r is so large as to require the fourth term of the series to be taken notice of, the circles which correspond to those radii will have their brightness diminished by a term of the form sin”39: they will therefore be most bright when sin 36 = 0, that is, where they are intersected by the rays drawn upon the screen as before mentioned; at points more remote from those rays the brightness will gradually diminish, and be least when sin 36 = 1, that is, at those points which lie exactly between them. 2. Let us now examine the image in the neighbourhood of the six rays; for this purpose 9 must be supposed small, and Z must be expressed in a series ascending by powers of 6. Upon this hypothesis we find |