cross the axis after refraction at the field-glass, and that where it actually crosses after emerging from the eye-glass, the angle of flexure, at each lens, is double of the original inclination of the pencil to the axis.

This simple system is however not applicable, as it is impossible to satisfy the condition necessary for perfect distinctness, much less that for destroying, as far as possible, the convexity of the field. These may however be very readily satisfied by employing two lenses of equal power, in each place, instead of one. The most proper forms of the lenses are those shown in Fig. 2, the field-glasses and the second eye-glass being of the meniscus form, and the first eye-glass equi-convex. I have found no sensible error arise from the substitution of plano-convex lenses for the meniscus glasses, which are difficult and expensive to form. Theory indicated a further flattening of the field, to be made by separating the eye-glasses a little, which requires the distance of the first eye-glass from the field-glasses, to be diminished by about half as much*; I cannot say however that I perceive any improvement arising from this alteration in practice, and as the field is quite flat enough with the eye-glasses in contact, and any further diminution of the apparent convexity, can be gained only by a sacrifice of distinctness, I cannot on the whole recommend it. I have not however yet had the instrument in a sufficiently perfect state of adjustment, in other respects, to be able to give a decided opinion on this point. This system, as it will easily be seen, gives a magnifying power of 3 to the eye-piece, so as to multiply, by that number, the power of the object-glass. It would be easy, if necessary, to produce a higher magnifying power, by employing lenses of

* The first eye-glass should in this case, be a little less curved on the lower surface and a little more on the upper, but it is hardly worth while to alter the form in practice.

shorter focal lengths, regard being had, in each case, to the proper condition of achromatism. Thus several different eyepieces might be inserted, at pleasure, into one tube, in the same manner as it is usual to vary the magnifying power of a telescope*. I have not yet tried the effect of this, but I suppose it may be necessary in applying the microscope to opaque objects, as the difficulty of illuminating them almost precludes the use of a powerful object-glass.

I do not pretend to give this as a perfect instrument-much less as one that will answer all purposes; but having tried it, in a very rough state, and with a moderate magnifying power, on various delicate test objects, all of which it shows very satisfactorily †, not excepting the striæ on the scales of the Podura, which Mr. Pritchard, the inventor of the diamond and sapphire lenses, says are only just discernible with the most perfect instruments, I see no reason to doubt that, when carefully executed, it will be found very effective, and that the naturalist may be furnished, at an expense not exceeding five or six guineas, with a microscope which will perform nearly all that can be expected from that instrument.

Fig. 3. represents the microscope, as I have directed it to be made by Mr. Cary.

* For example, if two equal plano-convex lenses having a joint focal length of half an inch, be placed at a distance of two inches and a half from the centre of the object-glass, and the eye-glasses be like those of the former eye-piece, but of focal lengths half an inch and one-third of an inch, the distance between the two pairs being in. the power of the object-glass will be multiplied about eight times.

18

+ Mr. R. Brown's active molecules may be very pleasantly observed with this microscope, with a power of about 360.

TRINITY COLlege,

April 23, 1830.

H. CODDINGTON.

XXI.

On the General Properties of Definite
Integrals.

By R. MURPHY, B.A.

FELLOW OF CAIUS COLLEGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read May 24, 1830.]

THE modern physical theories, particularly those on the propagation of heat, and the distribution of electricity, attach a new importance to the subject of definite integrals; being used in the former case as the simplest means of integrating the partial equations which express the variations of temperature in bodies, and in the latter case entering implicitly the equations for the equilibrium and motion of electricity developed on the surfaces of bodies-in many problems of this nature the form of the function under the sign of definite integration is unknown, and the resolution of equations in which terms of this nature are found, is generally attended with very considerable difficulty it is obvious that considerable advantages might in these and other researches be expected to result from the study of the general properties of definite integrals, that is, such properties as hold true whatever be the function under the sign of integration.