JuLy 20, 1899] 
NATURE 
27h 
logical point of view. The book is invaluable to all in- 
terested in the natural history of Africa, and is especially 
important as indicating the number of game animals to 
be met with in British territories and dependencies. 
R. 
An Introduction to the Carbon Compounds. By R. H. 
Adie, M.A., B.Sc. Pp. vili+ 90. (London: W. B. 
Clive.) 
WITHIN the brief compass of this work the author aims 
at introducing the student to some of the chief groups of 
the carbon compounds, as represented by familiar sub- 
stances, and at the same time at providing a series of 
experiments to illustrate the properties and reactions of 
these compounds. Thus the subject of the hydrocarbons 
is developed from an examination of the properties of 
coal gas, which leads to the study of marsh gas, ethane, 
olefine, acetylene and benzene. A feature of the book is 
that aromatic compounds are described along with fatty 
derivatives belonging to the same group, phenol along 
with alcohol, benzoic and salicylic acids along with acetic 
acid, aniline along with ethylamine, &c. This arrange- 
ment of the matter produces, no doubt owing to the 
severe compression, a somewhat disconnected effect, as 
it in many cases prevents a complete and logical discus- 
sion of the constitution of the compounds which are 
mentioned. This renders the book less suitable for 
absolute beginners than for students who have already a 
slight elementary acquaintance with the subject, and to 
these it cannot fail to afford valuable assistance. The 
experiments are on the whole well selected, but they are 
conducted on purely qualitative lines, no attention being 
paid to that important factor—the yield. AVE 
LETTER TO THE EDITOR. 
[Zhe Editor does not hold himself responsible for opinions ex- 
pressed by his correspondents. Netther can he undertake 
to return, or to correspond with the writers of, rejected 
manuscripts intended for this or any other part of NATURE. 
No notice ts taken of anonymous communications. | 
On the Deduction of Increase-Rates from Physical 
and other Tables. 
Pror. Perry has called my attention to a want which some- 
times arises in making practical deductions from tables. Take 
the following example. 
oc: p Ap A% «| Asp 
90 1463 ae | 
95 gifmamntz6s aa 49 8 
100 2116 408 57 
105 | 2524 470 62 : 
110 Bee) | 40 70 8 
115 Robt || . 618 78 
120 4152 
The table gives in the second column the pressure of steam 
for the temperatures stated in the first column, which proceed 
by equal steps of 5°. The question is, how best to derive from 
these data the value of ae at one of the stated temperatures, 
say 105”. 
The column A¢ gives the differences between consecutive 
values of 7. The column A*/ gives the differences between 
consecutive values of Af, and so on. The third differences A%/ 
exhibit so much irregularity that it is not worth while to proceed 
to fourth differences. 
It is obvious that the required result is greater than 1 of 408, 
and less than 4 of 470. Half the sum of these two is a fair 
first approximation. Closer approximations can be obtained by 
means of the numbers printed in large type. Let the down- 
ward sloping series 470, 70, 8 be called d@, dy d3, and the 
upward sloping series 408, 57, 8 be called 2, % 23. Also let 
the common difference 5° be denoted by 4. 
NO. 1551, VOL. 60] 
é ae ap. - 
It is known to mathematicians that ae is theoretically equal) 
to d,-4 d,+ 3 d,—&c., and also to #,+4 t+ 4 w3+&c., both 
series being supposed to be continued till we reach an order of 
differences that vanishes. 
In physical tables, usually no column of differences vanishes. 
exactly, and the two series will not exactly agree. The question 
is, how to get the best practical approximation out of them. 
The most obvious plan is to add them, and write 
d, 
2h f= (d, + uy) — (do — ta) + §(d; +g) — &e., 
then to take the first bracketed expression, the first two, the 
first three, &c., as first, second, third, &c., approximations. But 
it will be found on trial, in the present instance and in most 
instances, that the second approximation so obtained is less. 
exact than the first. 
I find, on looking into the matter strictly, that the proper 
second approximation is 
2h P= (dh +14) —¥ (dy ms). 
This equation would be exact if were capable of being 
expressed in the form 
p=Aée+ Be? + Co? + Det. 
As applied to the example before us, it gives 87°7 as the 
value of 2% <, and 8°77 as the value of This is as close an. 
ig 
approximation as is warranted by the data. The first approxim- 
ation (a +72) is 87°6. 
The two series ad, —4a, + &c., and #, +42 + &c., carried 
each to three terms, give respectively 87°53 and 87°83. 
The proper third approximation, which would be exact for 
p= A0 + Be? + Ce + Des + Eo® + Fe’, 
is 
2h 2, Bee, Ye 3s (d= ata ie (ae tea) 
Another requisite is to determine ge When the fourth order 
of differences vanishes, I find that @, —, is the accurate value of 
ieee In the present instance this gives 
at 62 = “48. 
de® 25 
The formulz most employed hitherto for this purpose are 
2p II 
h? —£ =, — dz + —d,- &e. 
Ee 
=Uy + tty + dtu, + ke. 
12 
: : . 2 . 62: 
which, if we include two terms of each, give respectively aE 
and os 
2 
When fourth and fifth differences are worthy of attention, the 
correction to be made for them consists in adding 
$(@y = 2) — q'(d@y + 2a) 
to the first approximation @, — 24. 
To take account of fifth and sixth differences, this correction 
must be supplemented by a further addition of 
ats (@y — Uy) — 35 (dq + Hy) + ayy — 243). 
Without occupying space by a detailed investigation, I may 
say that my plan of procedure is first to write down (by Taylor’s- 
theorem) the expansions for the first differences in ascending 
powers of #; then so to combine them in pairs by subtraction 
as to eliminate all even powers; then to eliminate /° from two 
of the resulting equations. This gives. 
an = (a, +2) —4(dy — tg) 
when /° is neglected. 
The next approximation is obtained by eliminating both 4* 
and /° from three of the equations. 
In the first operation for deducing a the pairs are combined 
ag~ 
by addition instead of subtraction, thus eliminating all od@ 
powers of 2. 
This gives dy — m= WL when /# is negligible. The suc- 
ceeding approximations are obtained by eliminating first 44 and 
then both 44 and 2°, J. D. EVERETT. 
