J. N. Stockwell— Researches on the Lunar Theory. 101 
t 
inequalities which do not disappear from the formulas of 
previous investigators by means of the same conditions. 
t might seem, however, that such large changes in the val- 
ues of the coefficients of some of the equations of the moon’s 
longitude, as my researches seem to indicate, would have a 
tendency to make the theory less accordant with observations 
than it is at present, since the present lunar tables represent the 
moon’s place within tolerably narrow limits. But a little con- 
sideration will show that such a conclusion would not neces- 
sarily follow. In order to illustrate this point, let us suppose 
that we have a perfect system of elements of the moon’s orbit 
together with a perfect theory of the perturbations. It would 
necessarily follow that the moon’s place could be perfectly pre- 
icted, and there would be no discordances between theory and 
observation. Suppose, now, that we omit a number of small 
though important equations from the computation of our ephem- 
eris, it would follow that there would be a series of residu- 
als between theory and observation. It is evident that these 
residuals would be perfectly represented by the omitted equa-° 
tions; but if the equations were considered as wholly lost, the 
theory would be in the same condition as though they had 
never been found; and we might seek to make up for the im- 
perfect theory by finding certain corrections to the elements by 
means of equations of condition between the variations of the 
elements and the observed residuals. In this way we might 
_ perhaps obtain a very good agreement between theory and ob- 
servations which extend over a limited interval of time,—the 
errors of the theory being partially compensated by the errors 
of the elements. But this close agreement between theory an 
but the residuals would furnish no information in regard to the 
nature of the equations to be applied in order to correct them, 
