69 
APPENDIX. 
As some readers might wish, to know the formula referred 
to in the text (sec. 27), I may state that it is, in its most 
general form — 
'2mv 2 =2'2.m /{^dx+ Ydy -\-7idz) + C (1) 
in which m denotes the mass of some one of the bodies or 
parts of the system, v its velocity, X, Y, and Z the resultants 
of the forces resolved along the axes of co-ordinates respec- 
tively, S the sum of like quantities (for instance, is the 
sum of the products of the masses multiplied each by the 
square of its velocity — called also the sum of the vires vivce), 
and C a constant quantity to be determined according to the 
value of 2mt i2 at some determinate position of the system. 
This equation takes different forms for different cases. In 
that of a planet revolving round the sun, where the mass of 
the planet may be taken as the unit, and the mass of the sun 
as immensely great, when compared with it, it is shown in 
II civ 
books on physical astronomy that X.dx + Ydy + Zdz = 
where /x is the sum of the attractions of the sun and planet, and 
r the distance of the latter from the former, or, more strictly, 
from their common centre of gravity, which is, qudm pr oxime, 
at the centre of the sun. Hence 2 f [Ydx + Ydy + Z dz) = 
“■ “t~ 0. The left-hand number of equation (1) is evidently in 
this case MY 2 + mx; 2 , where M and m are the masses of the sun 
and planet respectively, and Y and v their respective veloci- 
ties round the common centre of gravity. Now, we know 
that the quantities of motion MY and mv are equal ; therefore 
TYIV 
Y= — . And if we suppose M =mn, n being a very large 
number (in the case of Jupiter, the largest of the planets, 
in which n is smallest, it is 1,048), this equation becomes 
V=-. Hence MY 3 -i - mv 2 =mv 2 ( 1 -4--V in which - may be ne- 
n \ rj n J 
glected without sensible error. Thus equation (1) becomes in 
the present case, m'u 3 =?^-f C. This is the kinetic energy of 
r 
the planet at the part of its orbit where its velocity is v, v being 
variable. If we take w = l, and suppose v f the velocity at 
