98 REPORT—1846. ‘ 
t 
be obtained for any one of the twenty-four unknown quantities, for ex- — 
ample, Ag!' depends entirely on its weight, that is to say, on the magni- 
tude of the coefficient with which this quantity remains in the equation — 
containing at the origin the terms ...... + [(c.Ag!!).(c.Ag'!)].Ag'!, after 
the elimination of the twenty-three others ; and that these weights, as is easily — 
shown, will only become sufficiently extensive when there are neither two 
nor more of the unknown quantities whose coefficients remain in a constant, 
or in a nearly constant relation in the whole series of primary equations, tri- 
butary to the final ones. Hence the examination in this same respect of the 
above expressiuns for Aw, Ay, .........++. Ai, will easily show that by reason 
of the similarity of the latitudes in which by far the greatest part of the ob- 
servations till now calculated have been effected, the solution of the final 
equations as hitherto obtained would give but a very trifling weight to almost 
all the corrections we are seeking for, and therefore be still without interest. 
Even the seventy elements that, according to date of observation, follow 
next to the 283 finished ones, and for which M. Petersen has also nearly 
accomplished the primary equations, will, by their contributing to the final 
ones, most sensibly improve them in respect of solubility. Indeed in full 
opposition to the now finished ones, these latter elements relate to points of 
a line rather northern than eastern in its direction, and extending from 57° 
latitude north to about the equator. It is therefore precisely those unknown ~ 
quantities whose coefficients have hitherto exhibited the least variations, or 
followed in their varying a course parallel with that of the coefficients of 
other unknown ones, that will vary the most, both relatively and absolutely 
speaking, in the set of primary equations next to be formed, and thus will 
add to the final equations just what is requisite for increasing the weight of 
each of the quantities sought for and thereby preparing their due separation. 
M. Petersen will at all events subjoin, in the course of the ensuing months, 
this next continuation of his present labour, which I shall then forward to the 
Association. 
It is plain, notwithstanding, that even then we shall not have reached the 
most favourable state which our fund of observations for the year 1829 would 
allow of attaining in the knowledge of the Gaussian constants for the same 
year. Indeed even then, by substituting, as could be effected through your 
further patronage, a full execution of the task to a but partial one, the places 
of observation contributing to the final equations may be trebled, and what 
is still more important, a considerable improvement be attained in their re- 
petition over the globe. 
I thought it my duty to submit to the Association my opinion of the 
benefits to be derived from the continuation of M. Petersen’s labours, leaving 
it to their decision whether they will consider it advisable to grant him their — 
further support in devoting himself entirely to the prosecution of his task. 
