DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
31,3 
inately on particular hypotheses. For instance, if the pendulum never deviates much 
from the vertical, ^ is always small, and sec (p— 1 = nearly ; introducing this value, 
and substituting for f, we have for the non-periodic parts of the above expressions 
a' h’ 
I 0 ,“ -\-b^ I 3 nab 
Hence the axes of the mean ellipse are constant ; also if T he the time of describing 
27r 1 
the ellipse, we shall have approximately T=— whence it follows that the mo- 
tion of the apse during this period will be or ^•2'^ nearly. 
This agrees with the statement of the Astronomer Royal*. The above approxima- 
tions would cease to be sufficiently accurate if h were very nearly equal to a, or the 
motion very nearly circular; but they apply to any other case, and in particular to 
that in which h is very small, or the motion nearly rectilinear. In the case of nearly 
circular motion, it would be necessary to develope sec (p in a series of powers of 
and the results would be applicable whether p were small or not. But I shall 
not pursue this subject further here. 
Section VI. — On the Transformation of Variahles. 
61 . The method of the variation of elements, theoretically considered, consists 
merely in a transformation of variables of a particular kind ; that kind namely, 
which leads to a new system of differential equations belonging to the same general 
class as the original system. But practically, the choice of variables is determined 
by the well-known considerations from which the method derives its name. 
It is the object of the present section to consider the general class of trans- 
formations of which the method in question is a particular, and not the only useful 
case. 
62 . Definition of Normal Trahsformations. 
Let li, I25 ••• ^1; ^25 ••• JJn be new variables connected with the original variables 
j?i, &c. 3/1, &c. by 2n equations (which may also involve t explicitly), such that each 
variable of either set may be considered as a function of the variables of the other 
set (with or without t). Let P be any function of li, la? ••• 3/1? 3/2? and / ; then 
if the equations connecting the old and new variables can be put in the form 
(73.) 
I propose to call the transformation normal. 
* Proceedings of the Royal Astronomical Society, vol. xi. p. 160. 
