514 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
[587 
But we have 
2 _ /docV ( c tyX 2 (dzX 2 
\dt) \c?£/ "\c££/ ’ 
dhi drx d 0 dd _ dA 
X Tt~ V №~Jt r dt~ 2 dt’ 
the foregoing equation may be written 
i. v 2 . ,d 2 . 
nr- in TLr 
if — ■in' r X = C — 2 Itfrrdr — 2 I typdp, 
d.v 2 . , d 2 A _ , dr _ , dp 
- 4>n -^rrz = — 2(f>r ^ - 2\Jrp r 
dt’ 
whence 
the required result. 
12. If x, y are the coordinates of a particle moving in piano under the action of 
a central force varying as (distance)~ 2 : write down the expressions of the coordinates 
x, y in terms of the time t and of four arbitrary constants: and (in case of disturbed 
motion) starting from the equations 
Bx = 0, By = 0, Bx' = dt, By' = ^ dt, 
{the notation to be explained), indicate the process of finding the variations of the 
constants in terms of (1) ^, (2) the derived functions of il in regard to the 
constants. 
We have 
f cosu — e \/(l—e 2 )sinu . 
x = a •{- cos m + —, — sin m 
11 — e cos u 1 — e cos u 
where 
l\/(l — e 2 ) sm u cos u — e 
y = a \ —- cos m — s sm m 
° 1 — e cos u 
n — e sin u = t 
an equation serving to express u in terms of t and the constants a, e, c; the fore 
going equations, therefore, in effect give x, y in terms of t and the four constants 
a, e, c, m. 
In the second part of the question, O is a given function of x, y, t, the differential 
coefficients ^ being the partial ones in regard to x, y respectively. The equation 
Bx = 0 signifies that the variation of x, in so far as it arises from the variation of 
the constants, is = 0; it in fact means 
dx da + dx de ^ dx dc + dx dm _ ^ 
da dt de dt dc dt dm dt