in Mathematical Reasoning. 357 
be sought after; out of these, by means of some peculiarity in 
the problem, we must select that individual, which more imme- 
diately satisfies the particular view of it which we have taken, 
and the other solutions must be explained if possible, by means 
of the data, from which we commenced the process; or should 
that be impossible, their entrance must be traced to some gene- 
ralization in signs, to which the language of the question was 
incapable of adapting itself. In the demonstration of the com- 
position of forces, given in the Mecanique Cceleste, which has 
sometimes been unjustly censured on account of its analytical 
nature, this does not appear to have been completely attended 
to. In the enquiry, to which I refer, x one of the forces is 
assumed equal to z#(@), where z is the resultant force, and ¢(@) 
some function of the angle between it and the force x, which 
function it is required to determine; by changing 2 into y 
and @ into 5-9 the two values of « and y are found to be 
2=29(6), y=2¢(7-8), 
and the equation 
th +y¥ = 2? 
is arrived at: this equation in fact amounts to 
[er + [e(§-9)] =1, 
which results* from it, by merely substituting for x and y, their 
values. 
* This equation is one of that class whose general solution I have ascertained, and it may 
be exhibited in either of the following forms 
oo os 
o() +o (5-8) 
o@=\/ 5+ (5-28) x (45-9), 
ZZ2 
