proper precautions being taken with regard to the introduction of constants. Perhaps the difficulties relating to the constants may be evaded by writing the solution in the form

−(n+1)

u=ks" (d(tana)) { E‹ß− a) x (COS x)2(n+1)ƒ1⁄2(a¬ß) x (cos x) −2(n+1)dx},

and then substituting — (n+1) for n, which does not alter the original equation, we have

-2n

x)TMdx}.

u=ke" (d(tan x))" {é16—** (cos x) −1ƒ1⁄2"-k× (cos x)1⁄2′′dx' } .

If a and ẞ are both zero, we have for the solution of

α

Let a=c and ẞ=-c, so that the original equation becomes

d
x)

-2cx

=kɛ°* (d (tan a))** ' {ƒé-2* (cos x) −2(n+1)ƒf¿22 (cos x)2′′dxdx'},

which contains the proper number of constants; as the constant which enters by reason of the first integration disappears by the subsequent differentiations.

d dy

This gives for the solution of

This solution will apply to LAPLACE's equation, if for c be written c

54 MR. HARGREAVE ON THE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS.

d

the symbol ε-2cdy denoting that, after the operations are performed, y-2cx must be

d

written in lieu of y; and the symbol y denoting that, after the further operations are performed, y+cx must be written in lieu of y.

The solution is simplified by considering the latter function alone as a partial solution, and completing the solution by changing the sign of c with a new arbitrary function. Now if in LAPLACE's equation

The solution of LAPLACE's equation, therefore, by this process assumes the form

in

The following brief investigation is more general in its results than that developed pages 50 and 51.

By applying the fundamental theorem to the linear equation

u=4(x—n).4(x — 2n)...Σ{(px.4(x−—n).4(x — 2n)...) ̄1↓x},

(where the sign of summation has reference to x, Ax being n,) we obtain the equation

and its symbolical solution

ε ̄"*u-(D)u=4x,

u=4(D—n).4(D—2n)...{。=~'_1((ø(D).¢(D—n).q(D — 2n)...) ̄'‡x)} ;

and reducing, we obtain for the solution of (12.)

u=Σεpnx (p(D).q(D-n).... (D-pn)) ̄'4x,

pnx

=(9(D+pn).q(D+(p− 1)n).... (D)) ̄1{¿ ̄ï12↓x}.

If *=y, and xy consist of powers of y, the above formula gives the solution in series of powers of y of the equation

Several equations of this form solved by Mr. BOOLE's general method, are given in the Philosophical Transactions for 1844, pp. 236-240.

III. On a new substance occurring in the Urine of a patient with Mollities Ossium. By HENRY BENCE JONES, M.A., F.R.S., Physician to St. George's Hospital.

Received February 25,-Read April 22, 1847.

ON the 1st of November 1845 I received from Dr. WATSON the following note, with a test tube containing a thick, yellow, semi-solid substance :-"The tube contains urine of very high specific gravity; when boiled it becomes highly opake; on the addition of nitric acid it effervesces, assumes a reddish hue, becomes quite clear, but, as it cools, assumes the consistence and appearance which you see: heat reliquifies it. What is it?"

A few hours afterwards a specimen of the same urine, passed by a grocer fortyseven years of age, who had been out of health for thirteen months, was sent to me by Dr. MACINTYRE. He being in attendance on the case with Dr. WATSON, had two days previously first observed the peculiar reactions of the urine.

The specimen of urine was slightly acid; specific gravity 10342; it contained a sediment consisting of crystalline phosphate of lime, oxalate of lime, and cylinders of fibrin. The urine became thick with heat from a deposit of phosphates, but cleared with a drop of acid. It gave no precipitate with an excess of nitric acid, unless left to stand, or unless heated and left to cool, when it became solid. This solid redissolved by heat, and again formed on cooling. Continued boiling with strong nitric acid evolved but little gas, and did not quickly hinder this reaction. Hydrochloric acid gave the same solid precipitate, soluble by heat. Strong acetic acid gave only a slight precipitate, which redissolved by heat. Caustic potash and sulphate of copper gave a splendid bright blue, clear liquid, passing over when heated to claret colour.

516.84 grains evaporated to dryness in vacuo over sulphuric acid, gave 48.37 grains solid residue =93.58 per 1000 urine.

November 3rd.-I received more urine from the same patient. It gave a greater sediment, consisting of urate of ammonia, and some amorphous phosphate of lime, and some coarse cylinders. Filtered, specific gravity=1043.2. The reactions were the same as before.

521.39 grains gave 65.85 grains solid residue = 126.30 per 1000 urine.

All the phosphoric acid was precipitated in combination with lime by adding chloride of calcium and ammonia. Phosphate of lime 5.68 per 1000 urine.

=

November 7th.-Water from the same patient contained some sediment of crystalline phosphate, some granular and laminar phosphate of lime: some coarse cylinders

of fibrin, very coarse. The urine was slightly alkaline from ammonia; it gave a plentiful precipitate with alcohol; coagulated firmly with heat, very perfectly with a drop or two of acetic acid. Ferro-prussiate of potash gave no immediate precipitate; in less than half an hour a considerable precipitate had formed, which was soluble in liq. potassæ.

November 8th.-I saw multitudes of phosphate of lime crystals: no cylinders: few octohedral crystals. The urine coagulated with heat even when rendered feebly alkaline by liq. potassæ or liq. ammoniæ: if these were in excess coagulation by heat did not take place.

November 9th.-The urine was loaded with urates. Not nearly so readily coagulable by heat as it had been the two previous days; it bore today brisk and prolonged boiling specific gravity=1037.2.

November 15th.-I saw the patient. He stated that he usually passed about 353 of urine daily.

November 18th.-The urine had a specific gravity =1039.6 and was acid. It contained much urate of ammonia, phosphate of lime, and oxalate of lime; and bore long-continued boiling without coagulating. The urine was filtered and the following analysis was made:

519-54 grains evaporated to dryness in vacuo over sulphuric acid, gave,—

56.78 grains of solid residue =109.28 per 1000 urine.

Ash =6.02 grains 11:58 per 1000 urine.

This ash dissolved in water, filtered, precipitated by nitrate of silver and nitric acid, gave

Fused chloride of silver 48.84 grains=1·99 grain ch. sod. =3·83 per 1000 urine. Sulphate of baryta =1·47 grain =1∙10 grain sulphate of potash =2·11 per 1000 urine. 519-62 grains of urine precipitated by alcohol

Precipitate =37·16 grains =71·51 per 1000 urine.

29.26 grains of this precipitate gave,

1-47 grain ash (=5.01 per cent.) =3.58 per 1000 urine.

519-62 grains treated with alcohol

=

Dissolved 1972 grains=37.93 per 1000. Ash.. = 417 grains = 8:03 per 1000. 1024-48 grains precipitated by strong acetic acid—

Uric acid=99 grain =96 per 1000 urine.

Filtered fluid precipitated by ammonia,-

Earthy phosphate 1.23 grain 1.20 per 1000.

January 6th.—I was given by Dr. MACINTYRE the water passed on the 27th, 29th and 30th of December, all strongly acid, containing much urate of ammonia, oxalate of lime and phosphate of lime.

In the water of December the 27th, I saw some dead spermatozoa. The urine when filtered had a specific gravity of 1031.3.

515.37 grains precipitated by chloride of calcium and ammonia. Phosphate of lime 4.15 grains 805 per 1000 urine.

1018-85 grains, precipitated by ammonia. Earthy phosphates=1·47 grain =1·44 per 1000.

In that of December 29th, the specific gravity when filtered was =1037·9. 518-69 grains, precipitated as above. Phosphate of lime=4.27 grains=8·24 per 1000. 1027.71 grains, precipitated as above. Earthy phosphate 1.62 grain =1·57 per 1000.

In that of December 30th, specific gravity=10427, filtered. No fibrinous cylinders. The urine remained acid for a month in an open vessel.

521.05 grains, precipitated as above.

per 1000.

Phosphate of lime =617 grains 11.85

717.04 grains, precipitated by ammonia. Earthy phosphate 123 grain =1·72 per 1000.