320 
MR. R, S. HEATH ON THE DYNAMICS OF 
Assume, therefore, 
X( x > y) 
y) 
(o 2 =b 
a) 3 =c 
XX, y) 
X 2 X, y) 
XX, y) 
X%( x > y) 
°>i=f 
u-o=9 
XX, y) 
X%X> y) 
XX, y) 
XzX> y) 
a> 6 = h 
XiX, y) 
AsO®* y) 
where in all these formulae x—nt-\-a and y is arbitrary. The coefficients a, b, c,f g, 
h, and n, a are arbitrary constants. If we substitute these values of oq, oq, . . . co 6 in 
the equations of motion they give the equations 
A 
1-B-C 
B 
1-0-A 
C 
1 —A—B 
1 —A 
B-G 
d I S, 
na dx\Y^ 
U2 
7 d ( S 2 
nJ) dx 
d 
dx \A 12 
d / Sr 
*>£[?)=¥ 
n f 
dx \S 
U2 
L — i 
A 
A, 
As 
Aa 
X 
Al 
As 
Aa' 
x 
A, 
As 
A 2 
A 
A. 
14 
\ S 
•A^ As 
A Sj 
A 2 $ 12 
—ag 
A A 
*A 2 As 
A A 4 
A.2 A.2 A.2 A.2 
1 —B 
C-A 
ng 
d [X 
dx \d 12 
—ca 
, S u d 7 
X X 
Sj ^ ^12 ^12 
-vr 
1 - c n f b A (h±\— a jj A ^—fa A A 
A-B dx \XJ~ As As JJ As As 
Now these relations are of the same form as the relations between the As, which 
have been proved above. We have only to choose the arbitrary constants so as to 
make these equations exactly the same as the relations between the As, and then we 
have got a solution of the equations of motion. 
58. Compare first the terms involving products ; in this way we find the relations 
j= 
C 1 C 13 
y= 
b 
9 
G ° Gh h ^ 
a 
c 3 c 7 - 
4 r 
