June 1, 1899] 
NATURE 
IOI 
the ‘‘sum of an infinite series.” When 7,+2.+ ... is the 
series, the terms being uniform functions of x, the sum of the 
series for any value of .x is the limit of the sequence of numbers 
Uy, My tty, Uy +Ug+ug, ... in each of which x has the given 
value ; the limit of the sum of the series when =0, is the result 
obtained by /ivs¢ summing the series for a finite value of x, and 
afterwards diminishing x without limit ; the sum of the series 
when «=0 is the result obtained by /s¢ substituting o for « in 
the functions 7, w%, ... and afterwards forming the limit of the 
sequence 7%, %,+%, ..... In the example in question, the re- 
sults thus obtained are 4m and o respectively. The results that 
can be obtained by summing the series to 7 terms, diminishing 
xindefinitely, increasing 7 indefinitely and keeping 7x finite, 
generally do not coincide either with the sum for «=o or with 
the limit of the sum for x=0, when these are different. Such 
results may, as I have pointed out in a previous letter, be useful 
for purposes of illustration, but they are quite beside the mark 
when it isa question either of the statement of Fourier’s theorem 
or of the sum of Fourier’s series. 
M. Poincaré, in his letter printed in NATURE for May 18, 
does not assert that the sum of the series can be obtained by 
allowing .» to approach zero and 7z to increase at the same time, 
in such a way that z* remains finite; but he states that Prof. 
Michelson is perfectly right in contending that the result of this 
process is indeterminate. So far as I am aware this contention 
has not been called in question in the course of the discussion. 
Oxford, May 19. A. E. H. Love. 
Bessel’s Functions. 
THE remarks of ‘*C. G. K.”’ (p. 74) concerning the defects of 
style which are frequently observed in the writings of scientific 
men, lead me to point out a grammatical error which is creeping 
into mathematical literature. I allude to the use of the incorrect 
phrase ‘* Bessel Functions ” in the place of ‘‘ Bessel’s Functions.” 
In certain cases the name of a person may be converted into 
an adjective by the addition of an appropriate termination, of 
which such words as A/zzabethan and Vectorzan are examples ; 
but to use the name itself (which is a noun) as an adjective, isa 
violation of one of the most elementary rules of grammar. 
When the conversion of a proper noun into an adjective would 
be cumbrous or inelegant, the only correct mode of expression 
is to use the gevztzve case. If, therefore, we reject such an 
adjective as ‘* Bessellian” on the ground of its inelegance, we 
must use the phrase ‘‘ Bessel’s Functions,” that is functions of 
Bessel. The absurdity and incorrectness of the phrase ‘‘ Bessel 
Functions” is at once seen by comparing it with such phrases 
as ‘* Green Theorem,” ‘‘ Chrystal Algebra,” ‘* Love Elasticity.” 
The correct use of the genitive case is a subject upon which 
considerable misapprehension has existed. Thus we find in the 
Prayer Book the phrase ‘‘ For Jesus Christ His sake,” instead 
of ** For Jesus Christ’s sake.”” The error arose from the fact 
that the compilers of the Prayer Book were ignorant that the 
*s is not a conception of the pronoun /zs, but is the old Teutonic 
genitive which still exists in most German languages. 
Fledborough Hall, Holyport, May 28. A. B. BASSET. 
“The Art of Topography.” 
In your issue of March 23 (No. 1534, vol. lix.) appears a 
review of ‘* Recherches sur les Instruments, les Méthodes et le 
dessin Topographiques, par le Colonel A. Laussedat,” signed by 
““T. H. H.” The review brought to my attention several points 
of interest upon which I beg leave to comment. 
Regarding planetable instruments, the reviewer says ‘‘ that 
* Russians and Americans’ use very complicated instruments.” 
Of the Russian instruments I have no knowledge, but this is 
certainly not true of the American. 
The U.S. Geological Survey makes use of the planetable 
to a greater extent than any and all other organisations in 
America, fully two hundred of these instruments being con- 
stantly in use. 
The instruments used are remarkable in simplicity and 
efficiency, are reasonably light, portable and accurate. The 
instruments are of a model designed by Mr. Willard D. 
Johnson, of the Survey, and are fully described on pages 79 to 
89 of Monograph xxii. of the U.S. Geological Survey, entitled 
“* Manual of Topographic Methods,” by Mr. Henry Gannett. 
This work also treats of the methods of accomplishing topo- 
graphic mapping by the Geological Survey. Mr. Gannett 
explains the use made of the planetable, and shows that all work 
is controlled by points, located by triangulation or other means 
NO. 1544, VOL. 60] 
dependent upon numerical measurements and carefully com- 
puted. The triangulation is carried on with eight-inch theo- 
dolites reading, by micrometer microscopes, to two seconds. 
The instructions to triangulators include the order that points 
must be selected and arranged so as to best control the area 
under survey, and that vee points at least should be located on 
each atlas sheet of the map. Since these sheets differ in area 
in different parts of the country, ranging from 1/16 of a square 
degree to a square degree, the distance between tnangulation 
stations necessarily varies considerably. 
After the primary triangulation points are located in an area, 
dependence upon the planetable is absolute for the ‘‘ secondary ” 
triangulation within that area, the control, both horizontal and 
vertical, is carried on by use of this instrument. If the surveyor 
using a planetable for graphic work starts from accurately 
located points with check point available, he very soon dis- 
covers any ‘‘accumulation of error,” in that it is impossible to 
make the several locations check one with another. 
In regard to the use of ‘‘continuous contours” to express 
relief, the ‘‘Commission of 1826” seems to have drawn the 
remarkable conclusion that for scales less than I : 10,000 this 
system is insufficient. 
The Geological Survey publishes topographic maps which 
vary in scale between I : 9600 and I : 250,000 (1 inch to 800 feet 
and 1 inch to 4 miles about, respectively), and on these maps 
the contour interval varies between 5 feet and 200 feet. The 
expression of relief is, I think, in these cases satisfactory, at 
least so far as giving accurate information is concerned ; the 
artistic effect is very good also, especially when the topographic 
features are large and the slopes steep, cliffs appearing as broad 
heavy lines where differentiation of the individual contours is 
impossible. 
About 1890, the use of mercurial barometers was abandoned 
by the Geological Survey, and trigonometric methods for obtain- 
ing heights were adopted. At the present time the primary 
heights are determined by spirit-levelling, from which elevations 
are carried in connection with the triangulation or by lines run 
with vertical angle readings and carefully measured distances. 
The use of the aneroid barometer is only allowed in inaccessible 
areas between the known elevations, and must be frequently 
checked. The experience of the writer in widely separated 
regions in the United States, in obtaining differences of eleva- 
tion with the aneroid, leads him to the conclusion that, as a 
rule, the instrument fails to record differences as accurately 
when carried from a higher to a lower region as it does when 
the change of elevation is in the opposite direction. Also, that 
an aneroid which has been used in a region of elevation of given 
range must be given time to accommodate itself, if it be required 
to do good work in a region of greater or less elevation than 
that in which it has been used. The principle and construction 
of the aneroid is such that it never can be accepted as an instru- 
ment of precision except within well-defined limits, with 
frequent comparison with known elevations. The Survey has 
in use several hundred aneroid barometers, but no confidence 
may be had in any one of them unless frequently checked, as 
stated. It will be seen that the methods now in use in America 
agree more closely with those practised by the British Govern- 
ment, at least so far as the Colonial surveys are concerned, 
than with any other of the European surveys. Re dele (Cr 
The Heating of the Anti-Kathode in X-Ray Work. 
SINCE the beginning of X-ray work the heating of the anti- 
kathode has caused great difficulty, and with the introduction 
of the Wehnelt interrupter it is even more important that this 
should be prevented. In other words, we all along have had 
more energy from the coil than could be utilised in the Crookes’ 
tube. Many workers like myself have tried to remedy this, 
and various plans have been adopted to keep the anti-kathode 
cool. It occurred to me that if we could get a piece of platinum, 
fused into the glass tube itself, to act as the anti-kathode, and 
placed opposite the kathode, this object might be attained. 
Such a tube, after many attempts, has at last been made ; and 
although the first experiments have only been successful in 
making small tubes, others of a larger size are at present being 
attempted. The advantage of this method will easily be seen, 
because the heating of the piece of platinum can be prevented 
by placing the whole tube in a fluid cooling mixture or other- 
wise. These tubes are difficult to make at present, but I 
possess one which has retained its vacuum for some weeks, 
179 Bath Street, Glasgow, May 28. J. MACINTYRE. 
