730 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
The corresponding term in the corresponding gothic-letter function will be 
KA cos (ant -f- /3f2( + £—k), where K is the fraction by which the tide is reduced and 
k is the alteration of phase. 
It appears, from the inspection of the five formulas (24-8), that there are tides of 
seven speeds, viz. : 2(n —/2), 2 n, 2(n-\-fL), w —2/2, n, 71+2/2, 2/2. 
The following schedule gives the symbols to be introduced for reduction of tide and 
alteration of phase or lag. 
Semi-diurnal. Diurnal. Fortnightly. 
A 
~'\ 
r~ ^ 
Slow. Sidereal. 
Fast. 
Slow. Sidereal. Fast. 
Speed . 
2(n— f2), 
2 n, 
2 (n+ f2,) 
n— 2/2, 
n, 
n + 2/2, 
2/2 
Fraction of equilibrium tide . 
F 
f 3 
Gi 
G 
g 2 
H 
Retardation of phase or lag. . 
2fj. 
2f 
24 
8i 
g 
g 2 
2h 
The gothic-letter functions may now at once be written down from (24-8). 
Thus, 
2 (3£ 3 —§^ 2 ) = F 1 tn- 4 e 2(x ~ e ) -2fi +F2rcW*- 2f +¥ 2 K i e 2(x+B) ~^ 
+ F 1 urA-^- 9)+ ^+F2 OT Ve-^ +2f +FoK 4 e-^ +9)+3f = . . . (32) 
-4*a*v=i =the same, with second line of opposite sign .... (33) 
21? 2:——GjOT 3 /^* -30- & l +Gtn , /c(<T7trr — KK)e x ~% +G ;J trr/c 3 e x+2<)_ ' g2 
— G 1 73’ 3 /<:e~ (x ~~ e)+gl + G<n-K:(nTU7 — KK)e -x+s +G 3 a57<: 3 e -(x+3e)+S2 . . . (34) 
2 mv- l = the same, with second line of opposite sign ..(35) 
|-^ 2 =|-2 OT!! TK.+H OT 3 K^- 21 *+H®Ve- Zfl+2h .(36) 
The fact that there is no factor of the same kind as H in the first pair of (36) results 
from the assumption that the tides due to the motion of the nodes of the orbit are 
the equilibrium tides unaltered in phase. 
The formulas for 2(X /3 -Y' 3 ), -4X'YV=T, 2Y'Z', 2X'Z' A / = I, i~Z' 3 are found by 
symmetry, by merely accenting all the symbols in the five formulas (24-8) for the 
M functions. In the use made of these formulas this accentuation will be deemed to 
be done. 
At present we shall not regard y as being accented, but in § 12 and in Part III. we 
shall ha^e to regard y as also accented. 
We now have to develop the several products of the X' functions multiplied by the 
% functions. 
