Soient la latitude d'un point A sur une ligne géodésique, Z l'azimut de la ligne en ce point, et S sa longueur, on a généralement :

P

B

S=ƒ($)
S=ƒ(Z)

Si et Z reçoivent les accroissements 4 et 4 Z, la série de Taylor

En pratique, on peut négliger les termes qui contiennent les puissances de 4 et 4 Z supérieures à la deuxième.

Pour trouver la valeur des coefficients différentiels, supposons d'abord que la ligne géodésique soit un arc de grand cercle ayant son centre à l'intersection de la normale A avec l'axe de la terre et nous ferons ensuite une correction, afin de rendre le résultat applicable à l'ellipsoïde.

Formons le triangle différentiel A PB dans lequel

Ces formules sont suffisamment approchées pour la pratique. Si on le désire, on peut les corriger pour l'eccentricité de la terre en substituant à 4 son expression en fonction de la différence de latitude sur l'ellipsoïde. Les azimuts sur la sphère ne diffèrent de ceux sur l'ellipsoïde que d'une quantité du troisième ordre qui est trop petite pour être prise en considération.

La différence de latitude sur l'ellipsoïde est égale à 4 multiplié par le rapport de la grande normale au rayon de courbure du méridien.

Effectuant la substitution et exprimant 4 et 4 Z en secondes d'arc, on trouve:

4 représente maintenant la différence de latitude sur l'ellipsoïde ou telle qu'observée.

Les deux valeurs de 4 S obtenues au moyen des formules ci-dessus doivent être combinées suivant leurs poids.

Soient a et les erreurs probables de la différence de latitude et de la différence d'azimut; on en déduit pour 4 S les erreurs probables suivantes :

Si on a a=ẞ, Z= 45° et = 45°, alors ab, c'est-à-dire qu'à la latitude de 45° et avec un azimut de 45°, les observations de latitude et celles d'azimut donnent une distance avec la même précision, si les erreurs probables d'observation sont égales.

On a supposé que la différence entre les résultats obtenus par les latitudes et par les azimuts, était dûe aux erreurs d'observation; en pratique, on trouvera souvent qu'il n'en est pas ainsi, et que le manque d'accord est dû aux déviations de la verticale. La question exige alors un traitement différent.

Soient Set 4 S, les valeurs obtenues pour la distance, et 4 S, la vraie valeur.

Si nous admettons que le manque d'accord soit causé par les déviations de la verticale, l'erreur de 4 S sera dûe à la déviation A en latitude, et celle de 4 S, sera dûe uniquement à la déviation en longitude. Il est, en effet, facile de voir qu'une petite variation de la latitude n'aurait pas d'effet appréciable sur 4 S

D'après la théorie des erreurs, on doit adopter pour 4 S, la valeur pour laquelle

est minimum

Soit A la positition obtenue par les azimuts pour le point à déterminer, et Ccelle obtenue par les latitudes. Traçons le parallèle C D et le méridien A D; on a pour tout point B sur la ligne A C

La position à adopter pour le point à déterminer est donc celle pour laquelle B D est minimum; c'est-à dire le pied de la perpendiculaire abaissée de D sur A C.

Les relations suivantes se déduisent facilement de la figure:

En combinant les résultats des observations de latitude et de celles d'azimut, l'observateur devra être guidé par les circonstances, et employer soit l'une soit l'autre méthode, suivant qu'il croira devoir attribuer le manque d'accord aux déviations de la verticale ou aux erreurs d'observation.

On the Application of Hydriodic Acid as a Blowpipe reagent.*

By E. HAANEL, PH. D.

(Read May 25th, 1883.)

I. Some years ago it occurred to me that I might remove the difficulty experienced by students in distinguishing the oxyd-coating on charcoal of bismuth, from the similar one of lead by converting these coatings into iodides. The method adopted then was to touch the coatings with a drop of strong hydriodic acid, and direct the blowpipe flame upon the charcoal just in front of the moistened spot. The heat of the blowpipe flame volatilized the respective iodides, which were deposited again upon the cooler parts of the charcoal, at a greater distance from the assay. The iodide of lead gave a magnificent canary-yellow coating, the bismuth a chocolate-brown, cadmium and antimony, when treated in a similar manner, a white and brick-red coating respectively. This has since been published.

In the extension of this method to other substances, I found that other iodides of very characteristic colours were formed. Many of these were, however, altogether too volatile to be deposited satisfactorily on the charcoal, charcoal being too poor a conductor of heat to lower the temperature of the vapours of the iodides in question sufficiently to permit of their condensation and consequent deposition as coatings.

In order to utilize to the fullest extent the value of hydriodic acid as a blowpipe re-agent, it became necessary to adopt a support which, on account of its better conductivity, would condense the various volatile iodides on its surface as coatings. The choice of the kind of support best suited was farther restricted by the following characteristics which a support, to prove entirely practical and satisfactory, should possess.

1. It must be cheap and easily made.

2. The surface of the support must be smooth and white, to bring out the colours of the coatings, uninfluenced by peculiarities of surface or admixture of tint of the support. 3. It must resist the heat of the blowpipe flame.

4. It must be of sufficiently porous texture to absorb the hydriodic acid, and supply it to the assay gradually and constantly during the progress of the operation.

After some reflection and experimentation, I finally adopted plaster of Paris casts in the form of narrow, thin tablets, as the support, and found that it possessed the above-mentioned characteristics in an eminent degree.

II. PREPARATION OF TABLETS.

The tablets are made by mixing plaster of Paris with water to a thin paste, and distributing it over a large glass plate to the thickness of one-eighth of an inch. While yet moist and before the plaster has fully set, the surface is grooved with a knife, guided by a

* With a supplementary note on the spectroscopic examination of the colours of coatings obtained by this method, by A. P. Coleman, Ph.D., communicated by Dr. E. Haanel. (See page 72).

Sec. III., 1883. 9.

ruler, into strips of the size required for the tablets, most conveniently 4 x 13 inches. When dry, the cast is removed from the glass and broken along the grooves, made as indicated, into tablets. The surface next to the glass, which is smooth and level, is used for the deposition of the iodides. A small cavity at one end, made with a charcoal borer, serves to fix the position of the assay.

III-PREPARATION OF HYDRIODIC ACID.

The hydriodic acid used in the investigation was made by suspending a quantity of iodine in 7 oz. of water, and allowing hydrogen sulphide to bubble through it until the liquid became clear. More iodine was then added. Hydrogen sulphide was again passed through the liquid, and this process repeated until 5 oz. of iodine had been converted into hydriodic acid. The acid retains for a time a surplus of hydrogen sulphide, which is rather an advantage than otherwise, since it prevents the decomposition of the acid, and in no wise interferes with the reactions. The instability of the acid is practically of little importance. Any dissolved or precipitated iodine, which may occur after some weeks exposure to light and atmosphere, unavoidable during its use, is readily reconverted by hydrogen sulphide into hydriodic acid.

IV. MANIPULATION.

The assay is placed on the tablet in the cavity provided for its reception. One or two drops of hydriodic acid are now allowed to fall upon it, which are at once absorbed by the tablet immediately around the assay. The tablet is now held in an inclined position in exactly the same manner as charcoal when used for the production of coatings, and the point of the oxydizing flame directed upon the assay. Certain oxyds, chlorides, bromides, sulphides, etc., are thus decomposed by the hot vapour of hydriodic acid rising from the tablet around the assay, and any volatile iodides formed are deposited upon the tablet around and beyond the assay as coatings.

V.-RESULTS.

It is proper here to state that the investigation of the different elements in accordance with the method just sketched is not yet completed. As far as it has been carried and is described in this paper, it has proven beyond expectation fruitful. Seventeen elements have furnished more or less brilliant coatings, readily distinguished from each other by tint, difference in degree of volatility and certain typical characteristics, not easily described, depending on the mode of their deposition. Since no one can be expected from a mere verbal description to form an idea of the decisiveness and sharpness of the distinction. of the different coatings, and the superiority of this method over others employed in blowpipe analysis to determine the elements in question, and since it is important to the analyst to have the different typical coatings for comparison, my friend, Dr. Coleman, has been kind enough, at my request, to reproduce these coatings by water-colour sketches, which are appended to this paper. It is only by comparing the actual coatings with these representations, that the skill exhibited in thus accurately copying them can be appreciated.

*The paper "On the application of Hydriodic Acid as a Blowpipe-reagent" was accompanied by a chart on which the coatings obtained were represented by water-colour sketches. On account of the cost of reproducing them by chromo-lithography it became necessary to select from the number presented those which were most characteristic and served best to illustrate the value and advantages of the method described in the paper.