*885] . Two Planets beyond Neptune. 5 
planet at present ? what is its mass ? what are the elements 
of its orbit ? 
7* “ If we might determine, at every spot, the variation 
of the disturbance due to the unknown mass, we might 
therefrom deduce the diredtion in which Uranus falls by the 
incessant adtion of the troubling body, and we would know 
the position of the latter. But the problem does not present 
itself in so simple a form. The numerical expressions of 
the perturbations might only be immediately concluded from 
the observations, if we rigorously knew the elements of the 
ellipsis described round the Sun by Uranus ; these elements 
are again not to be exadtly determined unless we know the 
quantity of each of the figures of a theory. They allow to 
treat different points of a question without an error with 
regard to one of these influencing the other. However, it 
does not appear to me that we might follow this road with 
perfedt safety for the whole work, when we do not know a 
single relation to be satisfied by the result obtained, and 
which might serve for their verification. The mutual de- 
pendence causes that if the work is not completely corredt 
it is necessarily wrong in every point. Or we comprehend 
that it is easier to escape this alternative than to the mani- 
fold chances of an isolated error.” 
“ Indeed after having treated all perturbations simulta- 
neously, let us return to the calculation of a single one by a 
diredt method ; its verification will require that of the whole 
work. But if, instead of thus controlling a single inequality, 
we determine successively each one by diredt calculation, 
and then the new results coincide with the first one, all kind 
of error becomes impossible. It is this double work which 
I consider necessary to impose on myself in this case.” 
From this it appears that, notwithstanding this double 
work, when the disturbances observed and calculated — not 
knowing “ a single relation to be satisfied by the results 
obtained ” — are ascribed to one mass instead of three, the 
verification of the whole work which is all error will be 
required, turning the negative or evidently false results into 
new hypotheses of the existence of one, two, or more masses, 
as the case may indicate, utilising the one result confirmed by 
observation, as either accidentally right or as indicating an 
organic law. This is what I shall endeavour to do, as far 
as ability and time permit. 
8. The following, from the final memoir, is indeed a return 
from the second to the first method with its dangers. “ The 
methods which, like the preceding, lead at once to determine 
the perturbations separately one from the other, are most 
