DECEMBER 31, 1896] 
NATURE 
(Q7 
-of expressing my rational algebraical fraction, say $x, as a 
residue, by taking the distinct roots of the denominator, say p, 
and writing the variable equal to pe®, and taking the residue with 
changed sign of Sp—" «—™pp*, we can find the coefficient of 
x" or (if we please to say so) of x*" in the above square, and 
obtain a superior and inferior limit to the same in terms of 
Ps % , 2; and if, as I exfect (or rather, I should say, 
hope) may be the case, these two limits do not include zero 
between them, the theorems (mine, and therefore ev abundantia 
Euler’s) will be apodictally established. 
The two limits in question will be algebraic functions of 
ps9, » - » 54, whereas the aésolute value of the coefficient 
included within these limits would require a knowledge of the 
residues of each of these numbers in respect to every other as a 
modulus, and of 2z in respect of each of them. Ina word, the 
limits will be algebraical, but the quantity limited is an alge- 
braical function of the mid-primes Z, 7,7, . . - @ 
J. J. SYLVESTER. 
Athenzeum Club, December 20. 
P.S.—The shortest way of stating my refinement on the 
-Goldbach-Euler theorem is as follows :—‘‘It is always possible 
to find two primes differing by less than any given number 
whose sum is equal to twice that number.” 
Another more instructive and slightly more stringent state- 
ment of the new theorem is as follows. Any number 7 being 
given, it is possible to find two primes whose sum is 27, and 
whose difference is less than 7, 2 — 1, 2 — 2, 2 — 3, according 
as 2 divided by 4 leaves the remainders 1, 0, — I, — 2 
respectively. 
Major MacMahon, to whom and to the Council of the 
Mathematical Society of London I owe my renewed interest in 
this subject, informs me that in a very old paper in the Phz/o- 
sophical Magazine 1 stated that I was in possession of ‘‘a 
subtle method, which I had communicated to Prof. Cayley,” of 
finding the number of solutions in positive integers of any 
number of linear equations in any number of variables. This 
method (never printed) must have been in essence identical with 
that which within the last month I have discovered and shall, 
I hope, shortly publish.—J. J. SYLVESTER. 
‘Telegraphy without Wires, and the Guarding of Coast 
Lines by Electric Cable. 
Ir appears from an article in Commerce, December 16, that 
Mr. W. H. Preece, ina lecture on ‘‘Telegraphy without Wires,” 
at Toynbee Hall, said, that from experiments at the Goodwin 
Lightship it had been found impossible to get a message on 
board, and ‘‘that the intervening sea-water performed much the 
same function as an iron plate,” I would like to call the attention 
of the readers of NATURE to my paper laid before the Royal 
Society of Edinburgh in January 1893, when it was shown that 
neither salt nor fresh water had any appreciable effect on the 
transmission of these electrical waves. Take this case—an iron 
steamer afloat above a cable lying on the sea-bottom. If the 
steamer have on board suitable apparatus, messages sent along 
the cable from a single Leclanche cell can be and have been 
read on board ship by ordinary sailors. If it is possible to so 
‘convey messages to a vessel not moored by an anchor, it is 
surely possible to do the same to a moored ship such as a light- 
ship. Mr. Preece’s failure at the ‘‘ Goodwin” is not due to 
the action of salt water, for, if electric vibrations work through 
salt water in the Firth of Forth, they will equally do so at the 
**Gcodwin.” 
One word as to Prof. Boase and Mr. Marconi’s systems. 
Although it may be impossible to say what system may be found 
best for the detection of the electric vibrations, there is one 
thing certain that it is needless refinement to try to send the 
vibrations for lighthouse work ten miles. The vibrations 
require to be sent only 600 feet, as it is possible to lay a cable 
guarding a stretch of fifty miles of coast, ten miles off the shore, 
in at most fifty fathoms of water, and send the vibrations along 
it, and whenever the ship comes within two hundred yards of 
the cable the detector on board would give the alarm. 
Further, the advantage of the cable system is great, as the 
vessel would know her exact distance off; whereas, by sending 
the vibrations from a point on shore, this would be impossible. 
CHARLES A. STEVENSON. 
84 George Street, Edinburgh, December 21. 
NO. 1418, VOL. 55] 
The Origin of the Stratus-Cloud, and Some Suggested 
Changes in the International Methods of Cloud- 
Measurement. 
IN his “‘ Instructions for Observing Clouds” (London, 1888, 
p- 12), Hon. Ralph Abercromby defines straws as ‘‘a thin 
uniform layer of cloud at a very low level.” and as an illustra- 
tion reproduces a photograph of a low sheet of cloud which he 
says is exceedingly characteristic of east winds in London. In 
his book ‘‘ Weather,” p. 48, he shows by a diagram that the 
position of the s¢razzs is in the south-west quadrant of the anti- 
cyclone. By carefully plotting the observations made at the 
Blue Hill Meteorological Observatory during the past ten years, 
I find that this type of cloud has the same position in the anti- 
cyclones on the eastern coast of the United States that Aber- 
cromby found for England. Moreover the continucus records, 
made by instruments lifted by kites at the Blue Hill Observatory, 
furnish a very evident explanation of its origin. In a number of 
cases the recording instruments were lifted into or through such 
clouds, and in every case the temperature and humidity rose 
suddenly as the thermograph entered and passed through the 
stratus-cloud. This rise of temperature is not shown when the 
thermograph is lifted into cumulus or nimbus clouds. Hence it is 
evident that the stratus described by Abercromby is found at the 
plane of meeting between a cold current and a warmer, damp 
current overflowing it. The cause of the stratus is undoubtedly 
the mixture between the two currents and the consequent con- 
densation of moisture in the warmer current. 
There is, however, another conception of stratus described by 
Prof. H. H. Hildebrandsson in his *‘ Classification des Nuages 
employée 4 Observatoire météorologique d’Upsala,” where he 
says: ‘‘One sees that the stratus of Howard is nothing but a 
fog ; at Upsala we designate also, under the name of stratus, 
fog lifted above the earth, and which exists ordinarily as isolated 
fragments at a slight distance above the ground.” In the 
Hildebrandsson-K6ppen-Neumayer cloud-atlas a picture of one 
of these isolated fragments is given above the name of s¢vatus ; 
and the primary definition of stratus given in large letters is 
“Lifted Fog.” 
These two definitions of stratus by Abercromby and Hilde- 
brandsson have apparently been taken as identical by their 
authors ; but I think the facts mentioned indicate that they have 
no more in common, either in origin or appearance, than have 
cirrus or cumulus. When the International Committee met at 
Upsala it recognised the inadequacy of the illustration of stratus 
given in the Hildebrandsson-Koppen-Neumayer atlas, and, like 
Abercromby, /zctwved stratus as a thin sheet of low cloud, but 
defined it as ** Lifted fog in a horizontal stratum.” This com- 
promise between two entirely different conceptions of stratus 
results in an absurdity. Lifted fog rarely or never forms ina 
horizontal stratum. Certainly, during ten years of daily observ- 
ations of clouds, I have not seen such a phenomenon, nor have 
I seen it described ly writers on the subject. Moreover, if 
lifted fog ever does form in a horizontal stratum, how can an 
observer know, when he sees a stratus, whether it is lifted fog 
or is a cloud formed by mixture? I trust at some future meeting 
of the International Committee this definition may be changed. 
Probably the authors of the definition will not object to the 
change, now that the observations with kites have thrown a new 
light on the origin of stratus. 
Another point to which I think the attention of those engaged 
in the international scheme of measuring the heights and 
velocities of clouds should be called, is the fact that measure- 
ments of cloud-heights by theodolites or photogrammeters give 
erroneous averages for certain forms of clouds. At Blue Hill 
Observatory, using every opportunity to measure the altitude of 
nimbus with theodolites, we find the average height by such 
measurements to be 2077 metres ; yet in our measurements of 
cloud-heights, made by sending kites into them, we find that on 
more than half the days when nimbus is present its base is 
at an altidude of less than 1000 metres, and usually less than 
500 metres. The average height determined from the kite- 
measurements is 497 metres, and by the angle above the horizon 
of the light reflected at night from the clouds over distant cities 
it is found to be $45 metres. Similar differences are found in 
the case of strato-cumulus. The reasons are that low clouds 
are so indefinite in outline, or they cover the sky with such a 
uniform veil, that they cannot be measured with theodolites or 
photogrammeters. It results that the clouds measured by 
theodolities are principally high clouds. On the other hand 
very high clouds cannot be measured with kites, and the average 
