fundamental principle of the doctrine of chance, the probability that both these errors happen together will be expressed by the product XY. If now we were to determine the values of x and y from the equations x+y=E and XY=maximum, we х y ought evidently to arrive at the equation : and since x a b and y are rational functions of the simplest order possible of

a,

x y

α

=

b

b and E, we ought to arrive at the equation without the intervention of roots, in other words by simple equations; or, which amounts to the same thing in effect, if there be several forms of X and Y that will fulfill the required condition we must choose the simplest possible, as having the greatest possible degree of probability.

"Let X', Y' be the logarithms of X and Y, to any base or modulus; and when XY = max. its logarithm X'+Y' = max. and therefore X' + Ÿ' = 0, which fluxional equation we may express by X" + Y"y=0; for as X' involves only the variable quantity x, its fluxion X' will evidently involve only the fluxion of ; in like manner the fluxion of Y' may be expressed by Y"; and from the equation X"a+Y"=0 we have X"-Y"y: but since x+y= E we have also + y = 0, and i ż = −ỷ, by which dividing the equation X"¿ = -Ÿ"ÿ, we obtain X"Y".

"Now this equation ought to be equivalent to

х y
; and
b

α

this circumstance is effected in the simplest manner possible, by

tity a' being either absolute, or some function of the distance a. "We have discovered, therefore, that the logarithm of the probability that the error x happens in the distance a is expressed by a' + X', and consequently the probability it

mx2

2a

=e

mx2

a' +

2a

self is X = eX' Such is the formula by which the probabilities of different errors may be compared, when the values of the determinate quantities e, a' and m are properly adjusted. If this probability of the error x be denoted.

by u, the ordinate of a curve to the abscissa x, we shall have

(a+12) probability.

u = e

which is the general equation of the curve of

"When only the maximum of probability is required, we have no need of the values of e, a' and m; it is proper, however, to observe that m must be negative. This is easily shown. The probability that the errors x, y, z, etc., happen in the dis

tances a, b, c, etc., is e

which is equal to e

(a'+ max2)

2a

'+b'+c', etc., + +

(a'+1

(c'+

etc.

c.)

mz2
2c

and this

quantity will evidently be a maximum or minimum as its index or logarithm is a maximum or minimum; that is, when

=

2

etc.,

etc., it is evident therefore that m must be

value

negative; and as we may for the case of maxima use any of it we please, we may put m = -2, and the probability of x

: e(ƒ3—x2) for the equation of the curve of probability; but if we suppose f = 0, the ordinates u will still be proportional to their former values, and we shall have u = e

is the simplest form of the equation expressing the nature of the curve of probability."

Immediately following the above general solution by Dr. Adrian there are given applications of this method to the following problems.

1. To find the most probable value of any quantity of which a number of direct measures are given.

2. To find a most probable position of a point in space.

3. To correct in the most probable manner the dead reckoning at sea.

4. To correct the bearing and distances of a field survey.

The article closes on p. 109 of the Analyst with the following: "I have applied the principles of this essay to the determination of the most probable value of the earth's ellipticity, &c., but want of room will not permit me to give the investigation at this time."

The investigations here alluded to were, however, long afterwards published, i. e., in 1817, in vol. i, new series, of the Transactions of the American Philosophical Society, and are given in two papers (Nos. IV and XXVIII) of that volume. The preceding note as well as the dates written on the manuscripts (which are still preserved by the Hon. G. B. Adrian of New Brunswick) show that these two investigations were completed in 1808.

The first of the papers here alluded to is entitled "Investigation of the figure of the earth and of the gravity in different latitudes," from which as printed in the Phil. Trans., we make the following extract:

"Having in the year 1808 discovered a general method of resolving several useful problems by ascertaining the highest degree of probability, when certainty cannot be found, I shall here apply that method to the determining of the earth's ellipticity, &c." The author's computation is based on the lengths of the seconds pendulum as given by Laplace (Mec. Cel., iii), and having stated the problem before him, he says: "This is accomplished by a rule published by the writer in the Analyst, in 1808." The resulting ellipticity (1) he shows to differ from that deduced by Laplace (4) because of numerical errors in the computation of the latter; having corrected these he deduces the ellipticity by Laplace's own method-showing that the two methods conduce to nearly the same result.

In

The second of the articles in the Phil. Trans., is entitled "A Research concerning the Mean Diameter of the Earth." this the author seeks the sphere which most nearly coincides in various specified peculiarities with the actual terrestrial spheroid; the diameter of this sphere he determines to be 79187 miles. This numerical result is based upon some earlier computations, the details of which are not given, but of which he says: "Having determined the most probable axis of the terrestrial spheroid from the measurements of a degree of the meridian by a method which I discovered several years ago and published in the Analyst, the resulting mean radius was found to be 3959-69 English miles."

The mathematical works published by Dr. Adrian are so rarely to be met with, that it was necessary to make these long extracts in order to establish the conclusion to which we have arrived, i. e., that we must credit Dr. Adrian with the independent invention and application of the most valuable arithmetical process that has been invoked to aid the progress of the exact sciences.

Washington, Feb. 22, 1871,

ART. LX. Contributions to Chemistry from the Laboratory of the Lawrence Scientific School. No. 15.—On some new Analytical Methods; by THOMAS M. CHATARD.

§ 1. On the Determination of Molybdic Acid as Plumbic Molybdate.

GREAT difficulty has always been experienced in determining molybdenum by any of the methods generally in use. The precipitation as mercurous molybdate, and the subsequent treatment, are both tedious and unsatisfactory, and an accurate determination as sulphide is almost impossible, the filtrate remaining blue even after repeatedly passing sulphydric acid and filtering.

The following method will, I hope, be found to give satisfaction, both as to ease of working and accuracy of results.

Add to the boiling solution of the molybdate, plumbic acetate in slight excess. Boil for a few minutes; the precipitate, at first milky, will become granular and will subside easily, leaving a perfectly clear supernatant liquid. Care must be taken in boiling, as the thick milky fluid is very apt to boil over. A ribbed filter is to be used and the precipitate is to be washed with hot water. The washing proceeds with great ease and thoroughness, and not the slightest milkiness should be apparent in the filtrate. The precipitate is dried at 100°, separated from the filter and ignited in a porcelain crucible.

The results were as follows:

1.0744 grm. Na, MoO, gave 1-9176 grm. PbMoO1 = 46·64 p. c. Mo.

4

The theoretical percentage of Mo in Na, MoO, is 46.60 if Mo 96, this being the number given by the most recent determinations.

=

The process seems therefore to give very good quantitative results, and is both easy and expeditious. The precipitated molybdate separates easily from the filter and can be heated to low redness without decomposition.

Analogy would seem to offer a good method for tungstic acid and experimental analyses were made for this purpose, but after repeated trials the process was finally abandoned, as the precipitate came down so finely divided that it passed through the filter. The precipitation seems, however, to be complete, and I have hopes that with closer filter paper greater success will be obtained.

Attempts were made to determine arsenic as arsenate of lead but without good quantitative results. On the other hand

I tried to determine lead as molybdate, tungstate, and arsenate, but without success, as the presence of an excess of the precipitant in such cases seems to exert an injurious effect, the filtrate or washings speedily becoming cloudy.

It may not be out of place to mention some work upon molybdenum undertaken some time ago, which, though unsuccessful, is not without interest. Various methods were devised of weighing the molybdenum as sulphide.

One way was to precipitate molybdic acid as mercurous molybdate in the usual manner. The suspended molybdate was boiled and sulphydric acid passed into the boiling liquid. The mercurous molybdate was decomposed and mercurous sulphide and molybdic sulphide were the results. The sulphides were thrown upon a filter and washed with cold sulphydric acid water, dried and ignited in a current of sulphydric acid, but this process failed to give satisfactory results.

Again, sodic molybdate was heated with four parts of dry sodic hyposulphite till all the free sulphur was driven off, leaving, according to theory, Na,SO, +MOS3. The mass was then digested with hot water, filtered and treated as before. When any of the sulphur was left, it was found that some of the molybdenum went into solution as sulphomolybdate of sodium. Even after adopting every precaution, enough was still dissolved to vitiate the analysis.

§ 2. On the Evaporation to dryness of Gelatinous Precipitates.

In a former paper* I called attention to the fact that many gelatinous precipitates, when evaporated to dryness, became very granular and easy to filter. I gave some examples then and have since tested several others.

Titanic Acid.-Rutile was fused with sodic disulphate, the titanic acid precipitated by ammonic hydrate and evaporated to dryness. It became quite sandy and washed with great rapidity and thoroughness.

Glucina. -Pure glucina was dissolved in chlorhydric acid, precipitated by ammonic hydrate and evaporated to dryness. In this case as glucina is soluble in ammoniacal salts, it was necessary to ignite the dry mass to expel these. This was done in the platinum dish in which the evaporation was carried The residual glucina was sandy and washed with great ease; zirconic, niobic and tantalic acids when thus treated gave results which were all that could be desired.

on.

Ceric, lanthanic and didymic oxalates when treated with sulphuric acid, and ammonic hydrate added, gave also perfectly satisfactory results.

*This Journal, vol. 1, p. 247.