ABSTRACT
Now, it is often necessary to determine the properties of functions of stochastic variables. For example, the astrophysical measurements on which the Laplace model of accumulated errors is based (as in §7-2) were used by Laplace in the further calculations he made in his work dealing with the asymptotic stability of the solar system. Given this, suppose we let Y (x, t) be some kind of stability measure dependent on the accumulated measurement error x(t) obtained from a set of astrophysical observations. We now expand Y (x(t +t), t +t) as a Taylor series about the point (x(t), t), in which case we will have
Y (x(t +t), t +t) = Y (x(t), t)+x(t)∂Y ∂x
+t ∂Y ∂t
+ 1 2 [x(t)]2 ∂
2Y
∂x2
+[x(t)t] ∂ 2Y
∂x∂t + 1
2 (t)2
∂2Y
∂t2 + 1
6 [x(t)]3 ∂
3Y
∂x3 + . . .
