579] 
ADDRESS DELIVERED BY THE PRESIDENT. 
177 
group by themselves. Every a posteriori test is founded on the supposition, that the 
hypothesis necessarily or probably implies that certain conditions must be satisfied by 
the asteroids or their orbits, viz. in the one case the conditions are those which follow 
necessarily and immediately from the hypothesis itself, in the other case those which 
are deducible from it by the principle of random distribution. The two principal 
hypotheses are that of Olbers, where the asteroids are supposed to be the fragments 
of a shattered single planet: and the hypothesis that they were formed by the breaking 
up of a ring of nebulous matter. On the first hypothesis the orbits of all the 
asteroids once intersected in a common point; the second affords no conclusion equally 
susceptible of an a posteriori test. 
But for a rigorous or probable test of either hypothesis, what is needed is rigorous 
expressions in terms of the time for the eccentricity, inclination, and longitudes of 
perihelion and node of each of the asteroids considered, or, what is the same thing, 
the computation of the secular variations of the quantities h, l, p, q, which replace 
these elements. The investigation is applied to those asteroids the elements of which 
were determined with sufficient accuracy, and the eccentricities and inclinations of 
which were sufficiently small (limit taken is 11°). And the backbone of the memoir 
is the investigation of the h, l, p, q, for twenty-five asteroids included between the 
numbers (1) and (40). In this calculation, as was clearly necessary, the action of the 
asteroids on the larger planets and on each other was neglected; the expressions for 
the h, l, p, q, of the larger planets are regarded as given—they are, in fact, taken 
from Le Verrier (as calculated by him before the discovery of Neptune, but afterwards 
dh 
partially extended to that planet). The effect is that the differential coefficients &c. 
are given each of them as a sum of sines or cosines of arguments varying with the 
time; and thus, although the calculation is sufficiently laborious, the process is not one 
of the extreme labour and difficulty which it is in the case of the larger planets. 
The resulting table of the h, l, p, q, of the twenty-five asteroids has, of course, a 
value quite independent of the theoretical part of the memoir. Of this it is sufficient 
to say here that the conclusion is on the whole against Olbers’s hypothesis. The 
subject is resumed, and more fully examined in a paper in the Astronomische Nachricliten T 
t. lviii. 
“ Investigation of the Distance of the Sun and of the Elements which depend 
upon it, from the Observations of Mars made during the Opposition of 1862, and 
from other Sources,” Washington Observations for 1865, Appendix II., pp. 1—29. The 
chief part of this valuable Memoir is occupied with a determination of the solar 
parallax by the discussion of the observations of Mars made in 1862 on the plan of 
Winnecke: three partial discussions had previously appeared, but these having been by 
comparisons of pairs of observations, one in each hemisphere, many observations in one 
hemisphere were lost by want of a corresponding observation in the other hemisphere; 
and out of a total of nearly 300 observations, only 125 were utilised. The idea is, 
the perturbations of the Earth and Mars being perfectly known for the period under 
consideration, every observation of the planet would lead rigorously to an equation of 
condition between its parallax, the six elements of its orbit, and the six elements of 
c. ix. 23