Aucust 24, 1899] 
NATURE 392 
The formula (ii.), taken to the fifth central difference, gives 
= 181273, 
ax 
the true value being 
du 4 
— = 18127°224. 
ax 
The inaccuracy in the ordinary formula is, of course, due to 
the fact that a table such as the above never gives the exact 
value of the function tabulated, but only the nearest integral 
multiple of a certain unit (in this case oor). If we denote this 
unit by p, each tabulated value differs from the true value by some 
quantity lying between —4p and +4p. It may be shown that 
this makes it possible for A—4A®+4A*°—4,% to differ from its 
true value by as much as 4;°p, while @)—4co cannot differ from 
its true value by more than }p. Hence this latter formula is 
more accurate than the ordinary one in the ratio of 64:9, or 
about 7:1, when fifth differences are negligible. When only 
seventh differences are negligible, the formula ay—%ty+ 356 1S” 
more accurate than the ordinary formula, in the ratio of 832: 55, 
or about 15:1. 
(3) The formule (ii.) and (iii.) give the first and second 
differential coefficients for the values of the “‘ argument ” shown 
in thetable. It is often more useful to have them for the zzfer- 
mediate values. This requires a modification of the method of 
central differences. Let us write 
(2; + 24) =V 
Ary =A) 
F(A22) + A220 4) =Ay 
[STs As 
&e. 
Thus for the interval from 5°2 to 5°3, in the above example, 
we have 
V Ay Ay A; Ag As 
1908044 19065 1909 189 19 9 
With this notation, it may be shown that, for any value of x 
from 0 to I, 
I-—2¥ 
t= {Ve = a,} 
x(1- I-2x 
a x) A-"gAs} 
x(1 =a?) (2—2) E20) ) 
4! ita 10 Shh 
a(I —4*) (4 -—«*) (3-2) I-24 
= 5! {40> 14 4; 
ome BRE she: S. « ts 5.) SK bWS)) 
Or, if we write x=4+0, then for values of @ from —4 to 
+4; 
yh 49=1V +0A}} 
I-46" 
~ "Ha {a+ $A} 
1 — 46") (9 - 46° 
vl 46°) (9 - 46°) 
vaA Nt 
{ ai+204 | 
: 00 4 o Gen) 
Differentiating this last expression twice with regard to 6, and 
putting @=o0 we find 
du 
2*. 4! 
3 5 
I . 
Taya Oa ayhet os og t © eens (Ve) 
au\ _ 5D), 3229. Bs 
Ales 2444+ 576046 — 32256048 + . Gears (vii.) 
Thus for y=5°25, in the above example, we find 
du _ 
fe 
19057°17, 
the true value being 
(4) The ormula (iv.) is useiul for constructing tables by 
means of interpolation. For halving the intervals in a table, it 
gives 
y= V —}Ao4+-3,A,- 9 a, +—35_ag— . 
s "EESe- bio24° 3276810 
NO. 1556, VOL. 60] 
on (viii. ) 
Similarly, for subdivision of the intervals into fifths, 
# =V— "3A, — ‘08A, + ‘008A, + ‘0144A, — ‘0008644; 
/ — *0029568A, + °00012672A, + °000642048A,— . 
#2=V — “1A, — "12A,+ "004A, + °0224A, — °000448A,; 
—*0046592A, + ‘00006656A, + ‘OOIOI8368As— ... 
ux3=V+ "IA, —"I2A,— ‘004A,+ Kc. 
us=V + “3A, — 08A,— ‘008A,+ Ke. ; 
(ix. ) 
the terms in w; and ws being the same as in we and 2, but 
with signs alternately alike and different ; and the sequence of 
signs in each case being... ++—-—++ ... The corre- 
sponding formulze for subdivision into tenths might be found = 
poe is simpler to subdivide into halves and then again into 
ths. 
When several differences have to be taken into account, the 
above method of direct calculation is less troublesome than the 
ordinary process of building up the table by calculation of the 
sub-differences. 
In the formulz (ix.) the terms due to V and A, have been 
given in the form V-—°3A,, V-—‘1A,,.. . ; but in practice 
these terms would be obtained by successive additions of ‘24 
to w%p, so that it is not necessary to calculate V. 
August 16. W. F. SHEPPARD. 
Apparent Dark Lightning Flashes. 
ON the evening of the 5th of the present month we were 
visited by a severe thunderstorm, which passed practically over 
this place. The lightning was very vivid and at times occurred 
at intervals of only a few seconds. In order to photograph 
some of the flashes I placed a camera on my window sill and 
exposed four films for consecutive periods of 15 minutes each. 
During the exposures I was observing the sky, and repeatedly 
found that after nearly each bright flash I could see distinctly a 
reversed image of each flash zz any part of the sky to which I 
turned my head. These apparent dark flashes, or rather the 
images on my retina, lasted for sometimes 5 to 10 seconds. 
At the time I wondered whether dark flashes had ever been 
noticed before, and thought that this phenomenon was not un- 
commonly observed, but seeing Lord Kelvin’s letter in your issue 
of August 10, I send this note in case it may prove of interest. 
Westgate-on-Sea, August13. WILLIAM J. S. Lockyer. 
Subjective Impressions due to Retinal Fatigue. 
IN reading the interesting optical experience as described by 
Lord Kelvin in NATURE of August 10, it occurred to me that a 
somewhat similar effect on the eye, as noticed by myself, might 
be of interest. 
Frequently late in the evening, and with a dull cloudy sky, 
I have seen my own figure, at least in part, apparently projected 
in gigantic form high up on the cloudy background. 
This happened in the following manner. Going to the door 
of the house, and standing there with the strong light from the 
lobby or hall lamp shining out upon the gravel-walk in front, I 
saw my figure in shadow strongly defined upon the illuminated 
pathway. On raising my eyes quickly to the sky, I there saw 
the same form marked out on the dark clouds, but in a lighter 
shade. 
The effect on the eye, as in Lord Kelvin’s experience, is- 
doubtless that of fatigue : in my experience, however, the form 
observed being very dark as compared with the illuminated. 
bagkground, I received the complementary impression of a. 
light-coloured figure on a dark background. 
The time during which this impression remained when look- 
ing at the clouds might be a couple of seconds. 
August 14. W. J. MIniar. 
Mathematics of the Spinning-Top. 
Ir should have been stated on p. 321 that, while 6, is the 
angle between HQ and HQ’ in Fig. 1, p. 347, the angle 
between HS and HS’is @. At the same time this opportunity 
is available for some corrections, for which the printers are not 
responsible. On p. 321 the values of sin @; and sh @, should be 
interchanged ; on p. 348, after equation (35), read ... ‘* MX is 
the harmonic mean of MT, MT’ and of Mm, Mv’, ...” 
August 12. A. G, GREENHILL. 
