552 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
remark may be repeated that, that part of it which is non-periodic will only cause a 
minute change in the mean figure of the spheroid which is negligeable, and the part 
which is periodic will cause small tides of about the same magnitude as those caused 
by the tangential stresses. With respect to the tangential stresses, it is ci priori 
possible that they may cause a continued distortion of the spheroid, and they will 
cause certain small tides, whose relative importance we have to estimate. 
The expressions for the tangential stresses, which we have found above in (1), are 
not linear, and therefore we must consider the phenomenon in its entirety, and must 
not seek to consider the precessional and mutational effects apart from the tidal effects. 
The whole bodily potential which acts on the earth is that due to the moon (of 
which the full expression is given in equation (3) of “ Precession”), together with that 
due to the earth’s diurnal rotation (being 1 (-§-—cos 2 #)) ; the whole may be called 
AS. The form of the surface cr is that due to the tides and to the non-periodic part 
a n* 
of the moon’s potential, together with that due to rotation—being (^ — cos' 3 6). 
Now if we form the effective potential cr (S — fj -), which determines the tangential 
stresses between cr and the mean sphere, we shall find that all except periodic terms 
disappear. This is so whether we suppose the earth’s axis to be oblique or not to the 
lunar orbit, and also if the sun be supposed to act. 
Then if we differentiate these and form the expressions 
” LU 
WCr - -rrl S — ! 
cc do \ 
o a d 
war - 
a sin 0d<j) 
we shall find that there are no non-periodic terms in the expression giving the tan¬ 
gential stress along the meridian; and that the only non-periodic terms which exist in 
the expression giving the tangential stress perpendicular to the meridian are precisely 
those whose effects have been already considered as causing secular distortion, and 
which have their maximum effect when the obliquity is zero. 
Hence the whole result must be— 
(1) A very minute change in the permanent or average figure of the globe; 
(2) The secular distortion already investigated ; 
(3) Small tides of the second order. 
The one question which is of interest is, therefore—Can these small tides be of any 
importance ? 
The sum of the moments of all the tangential stresses which result from the above 
expressions, about a pair of axes in the equator, one 90° removed from the moon’s 
meridian and the other in the moon’s meridian, together give rise to the precessional 
and nutational couples. 
Hence it follows that part of the tangential stresses form a non-equibrating system of 
forces acting on the sphere’s surface. In order to find the distorting effects on the globe, 
