JuLy 13, 1899] 
experience know how laborious is the solution of twenty 
simultaneous equations. Prof. Brown estimates the solution 
of the equations at half the labour of obtaining them, in addition 
u =e 
E | 4 | = 
e | ee fe vie e 
Fst n = uv 
2 | 3 Argument. a Fi | gf o¢ EE 
8 & Bo) Cetra oO —-- 
nook: Pe cc. || MOREE: 
BS ne mo 
, : fo) 13 | 206265 | 070002 
5 y +/ 18 | 17000 2 
3 ée +/ | 21 | 350 ot 
4 a D | 9 80 | = 9°05 
5 is F II goo0o ool 
3 2 +2/ 21a 240 3 
7 Po oO It 340 3 
8 ee! ERCP) 20) | ee 4 
9 ee! +(/-7) 22) | 100 4 
10 2 B27 18 6 06 
II é? o 10 | = 6 
12 Be +2F 20 Aieey | || Cher 
3 ye) Cc II 400 o"4 
14 | ea D+/ 19 12 oe 
15 ea Dre | 20 | 14 ot 
16,.| . ° 9 ores | ee 
7 ae F+/ 10 15 0°06 
18 ye BZ, di, | 45 0°06 
19 hel Bays | / 10 u exer 
20 he’ F-/ II oe or 
oh ke D+F 10 | 4 oz 
22 é +37 y/ ut gl 
23 ee #/ 18 uw ed 
oa | ae +(2/+7) | 17 Sil 4 
25 ee’ £(2/-/’) | 18 3 4 
26 ee’ #2 Ne) 8 4 
27 ee’? + (2+ 22’) 16 | 5 oe 
28 | ceemmee(2—27) | 15 |. (2g 06 
2 ee’? 7 17 a BiG 
30 3 +37 |) es 03 O‘OL 
3 23 i | 16 OES |, 0:1 
32 ek? + (7+ 2F) U5) 4 Riga ot 
33 eke +(/-2F) 17 30 men i4 
34 ek? ari 16 I4 o"4 
35 ee | e(24+2F) | 15 | & OT 
36 Cent (20-2) 16 | z O57 
37 eR +1 We zOnny 4 o7 
38 |. ce eer | || | 18 eae’ ® 
39 | SD | 7 el 
40 celGee (2 2)) | 16 oe u 
4r | coMmeD (2-7) | 16) | » Ramey or 
42 | ea D+2/ 15 | Sean 0102 
43 | ¢?qm D eee | Ors | ie 
44 | Pa D+2F 16 O75 | OF 
45 ka D | 8 3 O;T 
46 | ea® +2 16 Oj), 9 O'T 
47 ea? aay if 16 | 0°002) 0'02 
48 a D 8 0'001) 0°03 
49 As 3F : es 
50 | 4am F ake aeae 2 
Br | ee F+2/ 10 10 I 
2 ke* | F-2/ TON) Ole rors 
33 ke | F 10 4mi0 i) 4 
54 hee! F+/+/ 10 5 Oe 
55 | zee’ iF \| 10 3 se 
56 | Acc e— | iL, || 2 or2 
57 | «Ae! B-/+/ II 4 O'2 
58 he'? F+2/ 10 o's 0°03 
59 he’? F-2/ Lo 008 | 03 
60 co | F | 10 Cite 2/05 
61 hea. D+F+/ 10 ol OZ 
62 hea D+F-/ 10 o'2 i 
63 | ka D+F+/ 10 | O72 | Riese 
64 | ke'a D+F-/ 10 | 0°5 | 07003 
65 hea? F 8 | 0'004| 0°06 
NO. 1550, VOL. 60] 
NATURE 
261 
to the fact that this portion of the work is peculiarly liable to 
numerical error. He may therefore be congratulated on having 
obtained an algebraical solution, reducing the operation of 
finding fresh terms to mere multiplication of series. The 
mathematical investigation is referred to as destined for public- 
ation elsewhere, and does not appear in the memoir. The 
underlying principle is that when in a differential equation of 
the zth order there are 2-1 integrals known, when the right- 
hand member of the equation is zero, then a particular integral 
in the general case can be obtained. In the lunar theory the 
differential equation is, in effect, of the fourth order, and 
three integrals are known, two representing the elliptic in- 
equality and the third a variation of the epoch. 
For forming the right-hand sides of later stages, the quotient 
of each set of terms by the variation terms is required. As 
divisions are troublesome, these quotients are the quantities 
sought in the first instance: the new set of terms can then be 
obtained by a multiplication. The quotients referred to are given 
algebraically as the sum of four products, each product being 
that of two series. It is inconvenient, in the numerical appli- 
cation of the above method, that small coefficients often appear 
as differences of comparatively large numbers. Dr. Brown 
gives as an example a case where a coefficient 2 arises as the 
sum of separate coefficients 
— 6418 + 6496 + 316 — 392 
from the above-mentioned four products. 
Terms of long period require a special treatment, but the 
general methods apply to the other terms of the group. The 
loss of accuracy is reduced to that due to the first, instead of 
the second, order of the small divisor. 
When the period is that of the elliptic inequality, a new part 
of the motion of the perigee has to be determined, Calling 
this new part ¢aje a new unknown term ¢"!(D + 72). #e ca/e appears, 
and is transposed to the right-hand side of the equation, so that 
the quantities A, which in other cases are completely known, 
now appear in the form B+ cj, where B, 4 are known. Dr. 
Brown has already shown in the first part how ce may be 
obtained before the coefficients of the inequalities are 
calculated. When this has been done, one of the equa- 
tions becomes redundant. Another is already redundant, 
until the meaning of the arbitrary constant denoting the 
ellipticity is defined with further precision, Dr. Brown 
defines the arbitrary constant so that €)—e€) =I to all orders; 
hence Ay=Aj. The other coefficients A,, A,’ consist of three 
parts, one proportional to ¢aje, and arising from the quantities 4, 
a second arising from the quantities B, and a third proportional 
to Aj. The two equations for which «=o then give a double 
determination of A», and furnish a check upon the numerical 
accuracy. Many of the quantities that occur in this arrangement 
of the computations are of service at subsequent stages. 
The treatment of the third coordinate follows the same lines, 
and only differs in being more simple. § 
The foregoing table exhibits the extent of the calculations 
already performed, and the results of the first part are for 
convenience included in it. 
The decrease of accuracy of the terms in the twenty-second 
and twenty-third groups is due to the period of one term 
approximating to the synodic period. Even in these cases, the 
coefficients are given to less than one-thousandth part of the 
least quantity that could be detected by observation. 
Pans 
INVESTIGATIONS OF DOUBLE CURRENTS” 
IN THE BOSPHORUS AND ELSEWHERE? 
AS my books and papers are published chiefly in the Russian 
language, they are not very well known in this country. 
A short account of some of my results may therefore not be with- 
out interest. I cannot, in the course of my address, make you 
familiar with all my works, and wish at the present moment 
only to draw your attention to the interesting phenomena of 
double currents in the Straits of Bosphorus, Gibraltar, Bab-el- 
Mandeb, Formosa, and La Pérouse. 
The Strait of Bosphorus joins the Black Sea and the Marmora 
Sea. The Black Sea water has in it—roughly speaking—half 
the quantity of salt found in the water of the Mediterranean. 
1 Abridged from a paper by Vice-Admiral S. Makaroff inthe Proceedings 
of the Royal Society of Edinburgh (vol. xxii. No. 4, 18¢9). 
