162 J. WN. Stockwell—Hill’s Supplement to Delaunay. 
6, we shall have the whole variation of R due to any variations 
in the codrdinates by means of the equation 
dR dR 
dk aX, dR dR dR 
= ee. au? Sh” ae © poset 2 
Jor+-(T )av+( _ )o0+( an) 90+ (FG ov (2) 
+(Sj) 99 
Now 4dr, dv and 40 denote the total perturbations of the codr- 
dinates; they must therefore be the sums of the perturbations 
arising from the action of the earth and sun. Putting there- 
fore d,r, 6,r, 8,v, 6,v, 6,0, 6,0 for the variations arising from the 
action of the earth and sun respectively, we shall have 
Orb rtor du=dvt+d 60=6,0+6,0 (3) 
If we substitute these values of dr, dv and 66 in equation (2), 
and retain only the terms depending on the products of the 
masses, we shall find, 
oR= (Fe Sr 4 Tb + (“ay) 88 | 
dR a aR ( | (4) 
+( ee ort “Te O00 + (Gq) OP J 
Now Mr. Hill has taken account of only the first three terms 
of this equation and has omitted all consideration of the last 
three. But if Mr. Hill’s values of d,r, 0,v, 6,0 are correct, these 
last three terms are of much more importance than the first 
three. Indeed it is one of the curiosities or rather paradoxes 
of the whole theory, that, if Mr. Hill’s value of 4,7; 0,v, 0,0 be 
substituted in the last equation the value of dR will be equal 
to —R,, and the whole disturbing function arising from the 
earth’s figure will vanish; at least as far as the terms depend- 
ing on the coefficient 8, of Mr. Hill’s work are concerned, and 
hence all the inequalities depending on that coefficient ought to 
vanish also. But if the values of dr, dv and 0,0 which I have 
determined, are substituted in the last three terms of equation 
(4) we get rational and consistent values, of the same order of 
magnitude as Mr. Hill has derived from the first three terms- 
This is a strong corroboration of the statement which I made 
at tbe Philadelphia meeting of the American Association i0 
September, namely, that the computed values of the lunar 
inequalities arising from the earth’s figure are erroneous. | 
is fundamental defect in the theory, to which I bave 
called attention, renders it entirely valueless as a scientific 
