NATURE

"elements" of lower atomic weight. Mr. Lockyer next
considers what will be the difference
phenomena, supposing that A contains B as an impurity
in the spectroscopic
and as a constituent.
spectrum of its own.
In both cases A will have a

impurity, will merely add its lines according to the
B, however, if
amount present, as we have above explained; whereas
present as an
if a constituent of A it will add its lines according to the
extent to which A is decomposed and B is set at liberty.
So that as the temperature increases the spectrum of A
will fade if A be a compound body, whereas

it will not

fade if A be a true element. Moreover, if A be a
compound body, the longest lines at one temperature
will not be the longest at another. The paper chiefly
deals with a discussion from this point of view of the
spectrum of calcium, iron, hydrogen, and lithium, as
observed at various temperatures; and it is shown that
precisely the kind of change which is to be expected on
the hypothesis of the non-elementary character of the
elements has been found to take place. Thus each of the
salts of calcium, so long as the temperature is below a
certain point, has a definite spectrum of its own, but
as the temperature is raised the spectrum of the
salt gradually dies out and very fine lines due to
the metal appear in the blue and violet portions of
the spectrum. At the temperature of the electric

arc the line in the blue is of great intensity, the
violet Hand K lines, as they are called, being still
thin; in the sun the Hand K lines are very thick, and
the line in the blue is of less intensity than either, and
much thinner than in the arc.

magnificent star photographs show that both the H and
K lines are present in the spectrum of a Aquilæ, the latter
Lastly, Dr. Huggins's
being, however, only about half the breadth of the former;
but that in the spectrum of a Lyræ and Sirius only the
H line of calcium is present. Similar evidence that these
different lines may represent different substances appears
to be afforded by Prof. Young's spectroscopic observations
of solar storms, he having seen the H line injected into
the chromosphere seventy-five times, the K line fifty times;
but the blue line, which is the all-important line of cal-
cium at the arc-temperature, was only injected thrice. In
the spectrum of iron two sets of three lines occur in the
region between H and G which are highly characteristic
of this metal. On comparing photographs of the solar
spectrum and of the spark taken between poles of iron,
the relative intensity of these triplets is seen to be abso-
lutely reversed; the lines barely visible in the spark
photograph being among the most prominent in that
of the solar spectrum, while the triplet, which is promi-
nent in the spark photograph, is represented by lines not
half so thick in the solar spectrum. Prof. Young has
observed during solar storms two very faint lines in the
iron spectrum near G injected thirty times into the chro-
mosphere, while one of the lines of the triplet was only
injected twice. These facts, Mr. Lockyer contends, at
once meet with a simple explanation if it be admitted
that the lines are produced by the vibration of several
distinct molecules.

"The lithium spectrum exhibits a series of changes with a rise of temperature precisely analogous to those observed in the case of calcium.

"In discussing the hydrogen spectrum, Mr. Lockyer adduces a number of most important and interesting facts and speculations. It is pointed out that the most refrangible line of hydrogen in the solar spectrum, h, is only seen in laboratory experiments when a very high temperature is employed, and that it was absent from the solar protuberances during the eclipse of 1875, although the other lines of hydrogen were photographed. This line also is coincident with the strongest line of indium as already recorded by Thalén, and may be photographed by volatilising indium in the electric arc, whereas palladium charged with hydrogen furnishes a photograph in

[Dec. 19, 1878

which none of the hydrogen lines are visible. By emline of hydrogen in the green is obtained without the blue ploying a very feeble spark at a very low pressure the Fand red lines which are seen when a stronger spark is case of calcium. In concluding this portion of his paper used, so that alterations undoubtedly take place in the Mr. Lockyer states that he has obtained evidence leading spectrum of hydrogen similar to those observed in the to the conclusion that the substance giving the nonreversed line in the chromosphere, which has been termed helium, and not form of matter, and also the substance giving the 1,474 previously identified with any known the other more complex than that which gives the F-line or coronal line, are really other forms of hydrogen, the one more simple than that which gives the h-line alone, alone.

and value, and that they will have much influence on the
"There can be no question that the facts brought
further development of spectrum analysis, to which he
forward by Mr. Lockyer are of the highest importance
narily dealt with by chemists that they will hesitate for
has already so largely contributed. But his arguments
the present to regard them as proof of the decomposition
are of a character so totally different from those ordi-
of the elements until either they are assured by competent
equally simple and probable hypothesis, or until what Mr.
physicists that they cannot be explained by any other
Lockyer has foreshadowed as taking place to such an
extent in other worlds has been realised beyond question
different ways at different temperatures, and thus of
yielding different spectra, just as a bell may give out
that the same molecule may be capable of vibrating in
or cavil in our own laboratories. It has been suggested
different notes when struck in different ways; and
although Mr. Lockyer has replied to this objection, it
fact, however, as Mr. Lockyer has pointed out, that the
change from the spectrum of a compound to the lowest
can scarcely be regarded as finally disposed of. The
temperature spectrum of its metallic element is of a
similar character to and even less in degree than the
change from the lowest temperature spectrum of the
metal to the spectra which it furnishes at higher tempe-
ratures does not appear to favour such an hypothesis,
and from the similarity in the phenomena it is difficult to
deny that in both cases decomposition does not equally
take place. Prof. Young's observations on the injection
of particular lines into the chromosphere during solar
storms are also difficult to reconcile with this view, and if
the conclusions drawn from previous researches are cor-
rect, it also does not account for the short line coin-
cidences which led Mr. Lockyer to his hypothesis.

regarded as elements are not necessarily simple bodies,
but merely substances which they are unable to decom-
"Chemists are careful to teach that what are at present
pose or which they have no special reason to regard as
compound bodies. The remarkable relations, both in
atomic weight and properties, existing between many of
the elements, tend, indeed, to show that they are related
hope that he will continue his researches in this direction,
and we trust that at no very distant time he may be able
in the manner Mr. Lockyer supposes.
even the most sceptical."
We sincerely
to bring forward evidence sufficiently clear to convince

NOTES

MR. J. M. WILSON, Mathematical Master at Rugby School, has been elected Master of Clifton College, in place of Dr. Percival, elected President of Trinity College, Oxford. Mr. probably than any head master for the teaching of science in Wilson has done very much for science at Rugby, and, although schools, still we hope that Mr. Wilson will prove a worthy sucDr. Percival is a difficult man to follow, and has done more

We are indebted to the kindness of Prof. Hayden for the following observation made by his party on the day of the last eclipse of the sun :-" Washington, D. C., November 18, 1878. Observation on total eclipse of the sun. Camp No. 4, Little Sandy, Wyoming Terr., July 29, 1878 :

Time of total eclipse as taken by A. D. Wilson
with telescope and theodolite

Mr. James Eccles, with smoked glass ...
W. H. Holmes, with smoked glass and telescope
of theodolite

Approximate latititude

longitude

m. sec.

2

27.5

2 26.0

2

27-0

42 8 25

109 9 52

The weather was very clear and quite calm during eclipse, but was preceded by a heavy wind."

At the recent meeting of the United States National Academy, we learn from Science News (the third number of which has reached us), Prof. Davidson incidentally announced that he hoped to be able to observe the intra-Mercurial planets without waiting for eclipses.

THE Japan Mail states that an astronomical observatory is to be established within the precincts of the Geographical Bureau of Tokio. The same journal also announces that telegraphic insulators, made at the village of Imari, in the province of Hizen, are of such good quality that they find large sale in

Europe.

A SCHEME for the extension of meteorological observations throughout Russia is now being discussed by the Russian government. The Central Meteorological Institution will be located at St. Petersburg. Provincial meteorological institutions will be created in connection with each Russian university, and these provincial institutions will have under their superintendence all meteorological stations of their respective districts, which stations will be largely increased in number.

We are informed that the Government of New South Wales

pose of the money intended to be spent on the rebuilding of the Tuileries Palace, for the completion of the isolation of the French national library.

GENERAL MYER has sent orders to Sergeant Jennings to leave in Europe the collections of the works published by the Signal Corps Central Office at Washington, exhibited at Paris. This valuable set has been deposited in the hands of M. Jarry, 46, Boulevard Magenta, Paris, one of the most active members of the Meteorological Society of France, who will be ready to give any information relating to them. It is expected that the series of publications of the United States Signal Corps will be exhibited next spring at the Palais de l'Industrie, in the scientific exhibition, and an improved weather indicator will be sent from Washington to be practically tested by predictions adapted to the peculiarities of the French climate.

WE notice the appearance of a Russian work, by M. Starchevsky - "Guide for the Russian in Central Asia"-being a collection of vocabularies of the languages, viz., Turkish (Djagatay, or Uzbeck), with an indication of the variations afforded by the Kashgar, Khiva, and Turkoman idioms; Kirghiz; Tartarian (Kazan and Orenburg idioms); Sart, or the town Uzbeck; and Tadjik (Bukhara idioms). The vocabularies also contain sketches of the grammars of these languages, the words being given in their Russian transcription. A second volume, containing the Russian, Uzbeck, Kirghiz, Tartar, and Tadjik part, will appear shortly.

It is probably now only a question of time for the electric light to become an everyday institution in our large towns. For several nights satisfactory experiments have been made on Holborn Viaduct, and, under most unfavourable atmospheric conditions, part of the Thames Embankment was illuminated the other evening.

IN Mr. A. S. Wilson's "Experiments on Turnip Seeds" (reprinted from the Transactions of the Botanical Society of "heavy and fine

has requested Mr. William Forster, Agent-General for the Edinburgh), he follows out Darwin's idea that

Colony, Prof. Liversidge, of the University of Sydney, and Mr. E. Combes, M.P., C.M.G., to collect information in the United Kingdom and on the Continent relative to the working of English and foreign technological museums and colleges, with a view to forming similar institutions in Sydney. A sum of money has been placed on the Estimates by the Government of

seeds tend to yield the finest plants." Mr. Wilson's conclusions are altogether in harmony with those of Darwin. The mean of a large number of experiments gave a product of 2 lbs. 7 oz. per seed in the case of large seeds, as against 2 lbs. It oz. in the case of small seeds.

In his just published report on the trade of Chinkiang, on the the Colony, to enable the Committee to purchase suitable speci Yangtsze-kiang, H.M.'s Consul mentions that there are rumours

mens.

We have no doubt that the Agent-General for New South Wales (3, Westminster Chambers, S. W.) will be extremely glad to receive from such institutions, or from any other source, reports or any information which would assist the Committee in its inquiries.

A CORRESPONDENT writes :-A singular project is on foot at Paris. M. Camille Flammarion, having published a number of articles to prove that the moon is not destitute of inhabitants, has been led to the idea of constructing a refracting telescope which will be powerful enough to see them. He is now busy organising a committee to collect the necessary funds.

An unexpected difficulty has stopped the construction of the works for the mounting of the great refractor at the Paris Observatory. This instrument is to be erected on the grounds which the Municipal Council had let to M. Leverrier for the nominal sum of 100 francs a year for a period of ninety-nine years. But there is a law that the Government funds cannot be spent for building purposes, except on grounds belonging to the Government; and the Municipal Council, who were so liberal as to the rent, want a high price for the ground.

M. LOCKROY, the editor of the Rappel, and representative in the French Chamber of Deputies, has introduced a bill to dis

that the coal, iron, and plumbago mines in that neighbourhood are to be opened without delay, under the auspices of the Viceroy, Li Hungchang. A British engineer in the employ of the Chinese Government has recently visited these mines and reported on them to the Viceroy Shên, at Nanking.

THE limit of permanent snow in the Caucasus is very variable, this mountain-group, of 156 geographical miles, lying between two seas and several steppe-regions, being consequently subject to the most opposite meteorological conditions. The average height of the snow-line on the Elburs, the highest point of the Caucasus, is 10,885 feet. The average height of the lower limits of the glaciers on the Elburs is 8,216 feet. The Kasbek is the centre of another region of glaciers and permanent snow-fields, in which the true situation of the snow-line is not yet accurately ascertained. A third region comprises the high ranges of the schist-system of Perikitel and Bogoz in Daghestan. The fourth region is the Schathdag, south of Daghestan, ten geographical miles from the Caspian Sea. In this last region the snow-limit reaches to 10,374 English feet above the sea-level. The average height of the snow-limit within the Caucasus is 10,600 feet. Local variations upwards and downwards are frequent, and there may be a difference of 3,200 feet between the maximum and the minimum. In the West Caucasian regions these conditions resemble those of South Europe; in the Eastern region they ❘ of Fishes," recounted an instance of apparent intelligence in a

offer an analogy with those of Asiatic mountain-groups, influenced by a continental climate. During the last few years some glaciers in the western half of the Caucasus have been retrograde, just as it has been observed with those of the Alps during these last fifteen years. In 1849 several Caucasian glaciers were evidently advancing through ancient forests. The glacier of the Kasbek, especially of Deftoraki, after having followed for some time the general retrograde movement, lately began to move forward again. Experience has proved, that, whenever this movement reaches a certain amount, the end of the glacier is broken away, and may cause serious catastrophes, as it did by stopping the chief military road from Tiflis along the Terek valley to Wladikawkas. Such observations are thus of high practical importance. The Deftoraki glacier may be paralleled with the Rosenthal and Vernagt glaciers of the North Tyrol, considering their variable periods and extreme alternations of progression and retrogression. For further details we refer to Dr. Abich's paper in Proceed. Imper. Geol. Instit. Vienna, March 5, 1878.

REPORTS come to Science News of a remarkable and very extensive series of caves discovered in Page county, Virginia, which, it is said, a scientific expedition will probably soon examine. Their great area, variety, curious formation, and natural ornaments, if the stories about them be true, are sufficient to place them among the wonders of the world.

We have received Decade I. of a "Prodromus of the Zoology of Victoria, or Figures and Descriptions of the Living Species of all Classes of the Victorian Indigenous Animals," by Prof. F. McCoy (London: Trübner and Co.). The plates in this first part are most beautifully coloured, and do infinite credit to the skill of the colony. There are three plates of snakes, three of fish, one of the giant earth-worm (Megascolides australis, McCoy), one devoted to three species of the day-moth, and the last two to two species of diurnal lepidoptera.

THE Commission for the Survey of New York State has been reappointed with an appropriation of 14,300 dollars a year. A large amount of preliminary, work has been done under Mr. J. S. Gardner.

ACCORDING to a note published by the Norddeutsche Allgemeine Zeitung, Berlin time will become the only one in use in the whole of the German Empire. The difference in time is thirty minutes minus on the coast of the German Ocean, and thirty-seven in advance in the eastern parts of Bavaria. This resolution has been fostered by a similar reform lately established in Sweden.

M. BARDOUX has re-organised the French Central Society of Agriculture, which will be styled the National Society of Agriculture. It will be composed of 82 ordinary members, to foreign members, 150 corresponding members in France and Algeria, and 50 foreign corresponding members. The President of the Republic is to be ex officio the patron of this Society, and the Minister of Agriculture and Trade the honorary president.

skate, observed by the author while officiating as curator of the Manchester Aquarium. On the occasion in question a morsel of food thrown into the tank fell directly in the angle formed by the glass front and the bottom. The skate, a large specimen, made several attempts to seize the food, but owing to the position of the mouth on the under-surface of the head, and the closeness of the food to the glass, he was unable to do so. He lay quite still for a while “ as though thinking;" then, suddenly raising himself in a slanting posture, the head inclined upwards and the under-surface of the body towards the food, the creature waved his broad expanse of fins and thereby created an upward current or wave in the water, which lifted the food from its position and carried it straight into his mouth.

At the Royal Institution a Course of Six Lectures (adapted to a juvenile auditory) on a Soap Bubble, will be given by Prof. Dewar, M.A., F.R.S., at 3 o'clock, on December 28, 31, January 2, 4, 7, 9 (1879).

On the New Jersey bank of the Delaware River, the skeleton of a man has recently been found buried in a standing position in a red sandy bluff overlooking the stream. A few inches below the surface the neck bones were found, and below these the rest of the skeleton except the bones of the hands and feet. The skull being wanting, it could not be determined whether the remains were those of an Indian or a white man; but the burial was peculiarly aboriginal. It was found that around the lower extremities were placed a number of large stones, showing traces of fire, together with charred wood; and there was no doubt that the bones of the feet had been burnt. Probably the man had been executed as a prisoner of war, being placed erect in the pit with a fire around his feet. He would appear to have been then buried, with the exception of his head. The skeleton when complete must have been six feet high.

THE new instalment of the Transactions of the Asiatic Society of Japan contains several papers of considerable interest. Mr. E. M. Satow, the Japanese Secretary of H.B.M.'s Legation at Tôkiô, who was one of the earliest labourers in the field of Japanese literature, contributes articles on the "Korean Potters in Satsuma," and the "Use of the Fire-Drill in Japan;" Mr. Aston one on "Hideyoshi's Invasion of Korea," and Mr. R. W. Atkinson "Notes on the Manufacture of Oshiroi" (white lead). There are also two contributions on earthquakes in Japan, and notes on some of the volcanic mountains of the empire.

PEAT FUEL is much used at Bremen and in other parts of North-western Germany, and increased attention has been paid of late years to its production and preparation. We learn from Consul Ward's Report that the vast tracts of marshy moors which are to be found in many parts of the German Empire, and more especially between the River Elbe and the Dutch frontier, are regarded as containing an immense amount of wealth in the form of peat fuel. With the view of developing

An earthquake was felt in Cologne and vicinity on December ❘ and improving the present means of producing and manufactur

Io at 11.35 Α.Μ. A similar commotion was felt in the provinces of Luxemburg and Namur, principally on the borders of the Ardennes forest; on the same day at 11.28 Brussels time. The duration of the shock was eight seconds, and it was accompanied by a well-defined noise, which awoke the inhabitants. On the following morning a meteor was observed at six o'clock in Alsatia, from Mulhausen to Colmar. A fire-ball travelling from north-west to south was seen exploding, exhibiting a display of natural fire-works. No noise was heard by any observer.

AT the last meeting of the Manchester Anglers' Association, Mr. F. J. Faraday, F.L.S., in reading a paper on the "Mind

ing this article, and of extending its consumption to districts where fuel is dear, an association was formed at Königsberg a few years ago, and was reconstituted last year at Schwerin. Their intention is to diffuse technical knowledge throughout the country with regard to peat production and manufacture.

THE Annual Report of the Belfast Naturalists' Field Club for 1876-77, contains a variety of matter, some of it of considerable scientific interest. There is a brief account of the

excursions in connection with the Club, embracing a good deal of topographical, antiquarian, and other information. At the winter session a variety of papers were read, some of which are reported at greater or less length. In connection with Mr.

THE Twentieth Report of the East Kent Natural History Society is, on the whole, satisfactory. It contains abstracts of several good papers read at the meetings. The Society has ninety-three members.

EXCAVATIONS in the "Dragon Cave" at Mixniz, Styria, have been already noticed (NATURE, vol. xviii. p. 618). The diggings made in June, 1878, by the Anthropological Society of Gratz, have brought to light some bones bearing indistinct marks of cutting and percussion. Above the stalagmitic layer over the hearth-stuff some bones were found, in loam, well preserved, but probably derived from an older site. They are greenish, and partly of an intense bluish-green tint; and Prof. C. Doelter finds that their composition approaches that of turquoise [boneturquoise?]. A full account by Prof. R. Hoernes will be found in the Proc. Imp. Geol. Instit. Vienna, August 31, 1878.

THE additions to the Zoological Society's Gardens during the past week include a Yellow Baboon (Cynocephalus babouin), from West Africa; two Ring-tailed Lemurs (Lemur catta), from Madagascar, presented by Mr. G. A. Shaw; a Green Monkey (Cercopithecus callitrichus), from West Africa, presented by Mr. J. Williams; a Common Fox (Canis vulpes), British, presented by Mr. Sutton Sharpe; a Woodcock (Scolopax rusticola), European, presented by Messrs. E. and W. H. Davis; a Common Swan (Cygnus olor), European, presented by Capt. Marx; a Ring-tailed Lemur (Lemur catta), from Madagascar, deposited; an Ocelot (Felis pardalis), from America; a Cereopsis Goose (Cercopsis nova-hollandia), from Australia; three Yellow-winged Blue Creepers (Careba cyanca), from South America, purchased.

ON HELIOTROPISM IN PLANTS

THE heliotropic phenomena in plants form the subject of a monograph by Herr Wiesner, the first part of which has been recently communicated to the Vienna Academy. The following outline from the Anzeiger of the Academy will give an idea of some of the fruits of the author's researches on this important subject.

tained that beyond a certain intensity no growth in length

occurs.

The third section treats of the relations between the refrangibility of the light rays, and the heliotropic effects. The experiments were made partly in the objective spectrum, partly in varieties of light, got by sending white light through coloured solutions.. It was proved that portions of plants very sensitive heliotropically, e.g., growing stems of Vicia sativa, undergo curvatures in all kinds of light, even in ultra-red and ultra-violet, with the exception of yellow. The maximum of the heliotropic force of light lies at the boundary between violet and ultra-violet; a second (smaller) in the ultra-red. From both maxima the power of the rays to produce heliotropism decreases gradually on to the yellow. Portions of plants little sensitive heliotropically, are no longer influenced by orange, or by red and green, or even (in the case of etiolated shoots of Salix alba) by ultra-red rays. The yellow rays quite stop the heliotropism, for, e.g., in pure red a quicker and stronger heliotropism occurs than in a light which gives yellow besides red.

In the fourth section experiments are described on the joint action of (positive and negative) heliotropism and (positive and negative) geotropism. It is here shown, inter alia, that, in the case of plants very sensitive heliotropically, the geotropism is, at the optimum of light-intensity, apparently extinguished, even in strongly geotropic organs; further, that in many organs (grow. ing stem of the pea), the heliotropic and geotropic powers of curvature disappear simultaneously; in others, however (stems of cress), the younger portions of the stem are more strongly heliotropic than the older, and the oldest after-growing portions of stem no longer show bendings in the light, but, through drawing action on one side (the heliotropic overhanging point of the stem), show apparently heliotropic curvatures chiefly due to growth, which are then counteracted by negative geotropism.

The arguments which go to prove that heliotropism is due to the phenomenon of unequal growth upon unequally-lit sides of an organ are set forth in the next section, and proof is offered that, for heliotropism as well as for growth in length, free oxygen is necessary.

The last chapter furnishes proof that the conditions for heliotropism remain constantly the same during its course, and coincide with the conditions for growth in length; further, that heliotropism (and the same holds good for geotropism) occurs as a phenomenon of induction. In this chapter it is also shown that when light induces heliotropism in an organ, a fresh heliotropic or geotropic induction meets with resistances, and can only come into action after extinction of action of the first; and that successive impulses of light and gravity, of which each by itself is capable of producing certain effe effects, do not have their actions added together together when when the effects that should be obtained separately are in the same direction, e.g., one and the same side of the organ is helped in its growth in length.

SOCIETIES AND ACADEMIES

LONDON

Royal Society, December 5.-" On a Machine for the Solution of Simultaneous Linear Equations," by Sir William Thomson. Let B1, B2, ... B, be n bodies each supported on a fixed axis (in practice each is to be supported on knife-edges like the beam of a balance).

Let P11, P21, P31, ... Pu be n pulleys, each pivoted on B1;

P12) P22, P399... PH2

112

P13, P239 P33) ... Риз

"

The first section treats of the history of the subject. In the second section the author studies the influence of light on heliotropism. The experiments were made in the light of a gas flame which burned under a constant pressure with a uniform intensity (luminous power = 6.5 spermaceti candles). The unit for the measurement of the light-intensity was the strength of this flame at the distance of one metre. It was found that in heliotropism three cardinal points of lightintensity are to be distinguished; an upper limit, a lower limit, and between the two an optimum of light intensity. Thus with decreasing intensity of light the strength of the heliotropic effect increases to a certain point, and beyond this point decreases. The lower limit referred to coincides with the lower limit of light-intensity for the stoppage of growth in length, while the upper limit does not coincide, or only occasionally coincides, with the upper limit of light-intensity for growth in length, for in the case of plants very sensitive heliotropically it lies higher, and in less sensitive plants lower, than the upper limit for growth in length. The mode of arrangement of the experiment in gas-light did not permit of determining in all cases the limiting values of the light-intensities; thus, for example, the upper limit for the heliotropism of etiolated shoots of Salix alba, and of the hypocotylous portion of the stem of Viscum album, and the lower limit for the heliotropism of the growing stem of vetch could ... from their zero positions ;

not be ascertained. The former lies above 400, the latter far below 0.008. The optima were found to lie between O'II (the growing stem of the pea) and 6.25 (etiolated shoots of Salix alba). Both with gas-light and with natural light it was ascer

B2;

C1, C2, C3... C, ben cords passing over the pulleys;

,, D1, P11, P12, P13, ... Pin, E1, be the course of C1;
D2, P21, P229 P231 ... P2n, E2,,,
C;

D1, E1, D2, E2... Dn, En, be fixed points;

11, 1, 13, ... In be the lengths of the cords between D1, E1, and D2, E2, ... and D., and En, along the courses stated above, when B1, B2, ... B, are in particular positions which will be called their zero positions; Let 4 + 1, Lo + lg, e, be their lengths between the same fixed points, when B1, B2 ... B, are turned through angles