3A3 + B3 = 12r3 cn3 u dn3 u + (1 + cn u)*,

= (1 + cn u) {12r3 (1 − cn u) (k'3 + k3 cn3 u) + (1 + cn u)3},

where the term in {} is in fact a perfect cube

= [1 + cn u + r2 (1 — cn u)]”.

(The last mentioned expression is in fact

=

· (1 + cn u)3 + r3 (1 − cn u) [3 (1 + cn u)2 + 3r3 (1 + cn u) (1 − cn u)

where the second term is

+r* (1− cn u)3],

= 12r2 (1 − cn u) [} (1 + cn3 u) + ‡r3 (1 — cn3u)],

or extracting the cube root y, = √1-a, has its foregoing value : and the differential expressions are then verified.

Suppose y=1, we have

that is

(m-1) (1+ cnu)2 + mr3 (1 — cn3 u) = 2r sn u dnu,

(m − 1)3 (1 + cn u)3 + 2m (m − 1) r2 (1 + cn u)2 (1 − cn u)

+ 3m3 (1 + cn u) (1 — cn u)2 = r2 (1 — cn u) {4 — 4k2 (1 — cn3 u)}

or observing that the right-hand side is

= r2 (1 − cn u) {(1 + cn u)2 + (1 − cn u)2 + r3 (1 + cn u) (1 − cn u)}

=

and multiplying by jr, the equation becomes

0 = } (m − 1)3 r2 (1 +cn u)3 + (2m3 — 2m + 1) (1 + cn u)2 (1 — cn u) +(m3 — 1) r2 (1 + cn u) (1 - cn u) - (1 — cn u);

viz. this is

0 = {}r2 (m2 − 1) (1 + cn u) − (1 − cn u)}',

as is immediately verified: hence writing fr=; value in question, y = 1,

(m3 − 1) (1 + cn u) — r2 (1 − cn u) = 0,

we have for the

which is one of the values of cn u derived from the equation x=0; but this equation x = 0 gives, not the foregoing equation, but

m® (1 + cn u)3 = {(1 + cn u) + r3 (1 — cn u)}3,

viz. the three values of cn u are the foregoing value and the two values obtained therefrom by changing m into wm and w'm respectively, w being an imaginary cube root of unity. In fact the curve +y=1, has at the point x = 0, y=1 an inflexion, the tangent being y= 1, so that this line meets the curve in the point counting three times; but the line ☛ = 0 meets the curve in the point, and besides in two imaginary points.

x=

(2) Continued observations on the state of an eye affected with a peculiar malformation. By Sir GEORGE BIDDELL AIRY, K.C.B., M.A., LL.D., D.C.L., Honorary Fellow of Trinity College, Astronomer Royal.

Nearly ten years have elapsed since I last reported to the Society the state of my eyes, as regards optical convergence of pencils of rays. I subjoin the results of an examination lately made, and I place them in series with those of preceding examinations, as serving to shew clearly the gradual change which takes place in the eye during a period exceeding 55 years.

I. Distance from the cornea of the left eye at which a luminous point presents the appearance of a nearly horizontal line.

In 1825, 3.5 inches; Reciprocal = 286; Difference = -'073.

In 1846, 47

......

;

.....

•213;

-'028.

-'006.

-.005.

II. Distance from the cornea of the left eye at which a luminous point presents the appearance of a nearly vertical line. In 1825, 60 inches; Reciprocal = 166; Difference = — '054.

In 1846, 8.9

In 1866, 10.6

112; ⚫094;

•100;

073.

-'018. + '006.

-'027.

III. Measure of the astigmatic power of the left eye at different epochs; estimated in each case by the difference of the reciprocals for the same date in the two preceding tables.

In 1825, Astigmatism = 120; Difference

In 1846,

In 1866,

In 1871,

In 1881,

101;

091;

079;

•101.

IV. Distance from the cornea of the right eye, at which

a luminous point is seen distinctly.

In 1846, 47 inches; Reciprocal = 213; Difference =

The image formed by the right eye is very perfect; although there are anomalous spots on the cornea or crystalline, and appearances which suggest a fault in the retina.

(3) On the mechanism of the renal secretion. By C. S. ROY, M.D.

The observations which formed the basis of this communication were made in part during the summer session of last year in conjunction with Professor Cohnheim, at the Leipzig Pathological Institute, and in part during the last autumn and winter in the physiological laboratory of this University. They were commenced with the hope of being able to elucidate a number of questions bearing upon the relation which exists between certain diseases of the kidney and cardiac hypertrophy.

It was first sought to obtain information upon this subject by investigating the manner and extent to which the action of the heart is affected by obstruction of the renal arteries and the other large branches of the aorta; the facts obtained by taking this line

of inquiry were not however of a kind fitted to throw light upon the problem which it was specially desired to solve.

It soon became evident that an investigation of the manner in which the renal secretion and circulation are normally regulated, and the relation which these bear to the regulating mechanism of the systemic circulation would be best fitted to supply information of the kind required.

The method employed was, to record graphically the changes in volume of one or both kidneys, while at the same time the changes in the blood-pressure in the aorta and the rapidity with which the urine was secreted were also recorded on the same revolving cylinder or, as continuous tracings, upon the paper of Ludwig's kymograph.

The method used for recording the changes in volume of the kidney is the same in principle as that of the plethysmograph. The kidney is enclosed in a rigid metal box, the arrangement being such that while the organ can freely expand or contract, and while the changes in volume are recorded by a lever writing with a light glass pen upon the kymograph paper, no obstruction is offered to the entrance and exit of blood by the renal vessels nor to the outflow of urine by the ureter. The kidney is surrounded by warm olive oil, which, however, is not in immediate contact with its surface, but is separated from it by a delicate flexible membrane of a kind which has already been referred to by the author in several of his published papers, and which prevents any escape of the oil by the side of the blood-vessels and other structures entering the hilus of the gland. It is impossible, consistently with the brevity desirable in a communication of this kind, and without the aid of a diagram, to describe in a satisfactory manner the exact arrangement of the parts of the instrument, and the reader is referred for the complete account of the method used to a paper which will shortly be published in conjunction with Professor Cohnheim giving an account of the first part of these observations. At present it must suffice to say that, when the instrument is in use, the kidney lies between two delicate, exceedingly flexible membranes, which apply themselves closely to its surface and to the surface of the structures entering the hilus of the organ, and that each of these membranes forms with each of the symmetrical halves of the box a chamber which is filled with oil and which communicates by a relatively wide flexible tube with the recording instrument.

The metal box is roughly kidney-shaped, and the two symmetrical halves, the edges of which meet in a plane corresponding to the long axis and the hilus of the kidney, are joined together by a hinge. Opposite the hinge each half of the box has a semicircular incision cut into it, and these together form a round hole through which pass the structures entering the hilus.

The recording instrument proper resembles in many respects an arrangement which was described by the author in a paper which appeared recently in the Journal of Physiology upon the form of the pulse-wave.

It need not be described here, and it must suffice to mention that this instrument permits of even rapid changes in the volume of the kidney being recorded without its producing the slightest change in the pressure of the fluid by which the kidney is surrounded. It allows also the numerical value of changes in volume being ascertained with exactness.

The rapidity of the flow of urine is followed with great convenience and accuracy by an arrangement by which each drop of urine, which flows from a narrow tube tied in the ureter, falling upon a light aluminium plate at the end of a well balanced lever, causes, by its impact, a momentary descent of the latter, which dips the point of a fine platinum wire into a mercury cup, closing thereby for an instant a galvanic current, and causing a mark to be made upon the paper of the kymograph by means of an electromagnetic marker.

Each tracing then shews, 1st. the changes in volume of the kidney, 2nd. the aortic blood-pressure, 3rd. the rapidity with which the urine is being secreted, 4th. the time (which was recorded by a seconds pendulum and an electro-magnetic marker in the ordinary way).

The operation is the same as that for nephrotomy, the kidney being reached from the lumbar aspect. It is cleared of all its connections leaving only intact the structures entering its hilus. It is then enclosed in the metal box which has previously been warmed, and the two compartments of which are now filled with warm olive oil which fills also the flexible tube connecting the box with the recording instrument. It need scarcely be added that the animal,-rabbit, cat or dog, in most cases the latter, was kept fully under the influence of ether, chloroform or morphia, or a combination of two of these, from the commencement to the end of the experiment.

Contrary to what might reasonably have been anticipated, the kidney continues to secrete urine of normal quality and in quantity more or less exactly the same as that of the intact organ of the other side, for many hours after it has been placed in the metal box.

The following brief and incomplete resumé may serve to indicate the nature of the principal facts arrived at by Professor Cohnheim and the author in Leipzig, or by the author independently in Dr M. Foster's laboratory.

1. The volume of the kidney varies with each pulse-wave and with the respiration curves of the blood-pressure;-the tracing obtained resembling closely that of the mercurial kymograph.