184 
useful in teaching elementary classes. 
I consider that the proper test of limiting equality 
is that the difference between two quantities should 
become (numerically) less than any assignable fraction 
of one of the qtiantities, however small, in other words 
that + -a<ae where e is any fraction of unity, however 
small (instead of x -a<e where e is any quantity, how- 
ever small), the present definition being assumed to hold 
good even if the two quantities vanish or become infinite 
at the limit. 
If this condition be accepted as the definition of 
limiting equality, the same condition will hold good 
for any multiples or submultiples, however large or 
small, of quantities which tend to limiting equality, 
and also to sums of such quantities ; thus if x, —a,<ay,e, 
¥_-—A,<ae, etc., then =v -Ya<eXa under all condi- 
tions. Such statements as dy =f’(x)dx are to be in- 
terpreted as statements of limiting equality according 
to this definition, and we arrive at a definition of an 
integral as the limit of a sum of products, which is 
applicable not only to integrals of functions of a 
single variable, but also to integrals taken over areas, 
volumes, and indeed any of the concrete magnitudes 
which commonly occur in problems on mechanics and 
physics. Roughly speaking, this definition may be 
worded somewhat as follows : 
Let x be any magnitude which can be divided into 
elements Av, however small, y a measure associated 
with it such that if y, and y, are the greatest and 
least values of y associated with any element Ax, 
y, and y, tend to limiting equality when the mag- 
nitude Ayx diminishes indefinitely. Then, since 
(Vax - y,4x)/y,Ax = (V2 - ¥1)/¥y, the products y,Av 
and y,4x also tend to limiting equality, and by the 
theorem for the limit of a sum, the sums of the 
products y,A¥ and y,Ax taken over all the elements 
also tend to limiting equality and their common 
limit is defined as the integral [{ydx. Any single pro- 
duct can be legitimately designated by ydx in any 
equation, provided that this equation is interpreted 
as a statement of limiting equality in accordance with 
the above definition. A subsequent proof is required 
to cover cases of discontinuity such as occur, e.g. 
when finding the volume integral of a function which 
changes by a finite amount in crossing a surface. 
In defining a differential coefficient and proving 
the formula for the differentiation of a product, I 
follow Fricke’s method to a great extent. Fricke, 
however, defines f’(¥) by putting x,>y* in {f(x,) 
~f(*)}+(%, -%), but I consider it preferable to con- 
sider the more general fraction {f(¥_) — f(¥,)} + (#2 — 4). 
If this fraction tends to a unique limit when x, and 
%, approach a common limit x by any process what- 
ever, this limit is defined as the differential coeffi- 
cient of f(¥). This condition covers the cases where 
either x, or *, is first put equal to x and the other 
variable becomes equal to x subsequently. 
: : G. H. Bryan. 
University College of North Wales, Bangor. 

Museums. 
Tue article in Nature of December 9 on ‘‘ A 
Suggested Royal Commission on Museums ”’ leads me 
to offer a few comments, based on recent experiences. 
It is trite to say that all museums are understaffed, 
but it may be worth while to point out some of the 
consequences of this condition. Being a student of 
wild bees (Apoidea), I have long been interested in 
the available collections of these insects. In 1920-21, 
I made a catalogue of all the species of bees in the 
NATURE 
limit the test appears scarcely to be necessary or | Cambridge. 
[FEBRUARY 10, 1923 
Returning to America I catalogued the 
bees in the U.S. National Museum and the American 
Museum of Natural History. One of my principal 
objects was to bring about ex¢hanges between these 
institutions, so I noted in most cases the size of the 
series. The authorities everywhere were extremely 
cordial to the exchange idea, and it was evident that 
if each museum would distribute its duplicates, which 
were often actually in the waypall would be greatly 
enriched, to the advantage of students on both sides 
of the Atlantic. Up to the present, it has been 
impossible to carry out the proposed plans, because 
the curators have been fully occupied in other ways. 
The prospects will necessarily remain unfavourable, 
so long as each man has many more duties than he 
can attend to. The staffs should be increased, and 
should include at least two types of men—those who 
are principally concerned with research and those 
who are primarily curators. The latter type, with a 
passion for collecting and arrangement, is not to be 
found everywhere, and is not produced by the 
universities. It involves, however, a high grade of 
ability, and should be zealously sought by heads of 
museums. 
In their zeal for economy, many will object to 
increasing museum staffs. They ought to consider 
the matter as they would a factory or other com- 
mercial plant. A great deal of capital, material and 
otherwise, has been put into our museums. With 
a moderate increase of funds they can be made to 
function far more efficiently and develop more 
rapidly. The public policy has too generally been 
like that of a man who had built a house, and decided 
that he could not afford a roof. In some cases sheer 
poverty may afford an excuse, but even the United 
States, with all its wealth, treats its National Museum 
in the most niggardly manner. The truth is, that in 
a democracy the public will is the driving force, and 
an ignorant public has no will. It is the duty of 
scientific men to carry on a campaign of publicity, 
which need not involve anything detrimental to their 
self-respect. 
One reform which I should much like to see at the 
British Museum (Natural History) is the establish- 
ment of a room of British entomology, with a special 
curator who made it his business to know the species 
of the country. As things are at present, the average 
collector is interested primarily in British species, but 
on going down to the Insect Room he has to appeal ~ 
for help to a world-specialist in some group, who is 
perhaps monographing a particular family of beetles. 
Any one with a conscience hates to take much of the 
time of such a man for his relatively insignificant 
matters, and the specialist himself probably does not 
know the British Staphylinide or weevils. By 
assembling the British series in one room, in charge of 
a special man, or preferably two or three, the work of 
the amateur would be greatly facilitated, and young 
naturalists would not be blighted in the bud by a 
sense of the trifling character of their pursuits. This 
is not a criticism of the existing curators, whose 
courtesy and good nature under stress have often 
caused me to marvel. 
Just to show the spirit of the place, I will relate a 
couple of amusing instances, which I myself witnessed. 
A man came to the department of insects with an 
account of a proposed patent for catching fleas by 
entangling their feet in the supposed perforations in 
diatoms. The nature of the markings on the siliceous 
framework of diatoms, and their relative size to the 
feet of a flea, were explained in all gravity and- 
kindness, and presumably the new flea-powder never. 
appeared on the market! Another day, a man came 
British Museum, and also listed those at Oxford and | to the department of geology with a clay model of an 
NO. 2780, VOL. 111] 
