322 
MR. R. S. HEATH OH THE DYNAMICS OF 
59. Again, comparing the other terms of the equations with the relations between 
the 6’a, we deduce six more relations 
eg _ 
CjC 13 nA 
a 
c 4 c 8 1 —B —C 
ah 
C 0 C 14 AB 
b “ 
^ 2^12 ^ h A 
¥ _ 
Cc,c,r, nC 
c 
^ 9 ^ 12 1 -A. I> 
1 
ll 
c o c n A) 
/ 
cy 9 C —B 
¥ 
C 2 C 11 b) 
g 
c o c 9 A C 
fg 
c 9 c u rc(l-C) 
h 
b A 
From the last three equations 
<m 2 , (l-BXl-C) 
J c* L (B —A)(A —C) 
, 7 * = c n S (1-C)(1-A) 
J c* (C—B)(B—A) 
¥ (A-C)(C-B)' 
These are all positive if A, C, B are in descending order of magnitude. 
b 
’ 9 
pletely in terms of A, B, C, and the c’s. If we substitute these values in the first 
three equations of the last set, we are left with three relations among the constants. 
Now the @’s implicitly involve the eight constants, p, q, r, v, w, n, a, and the other 
variable y, which is perfectly arbitrary. Among these quantities we have seen that 
there are four relations imposed by the form of the equations of motion. Hence 
there are four arbitrary constants left to express the initial conditions. But to do 
this completely, we should require six constants. Thus the solution is not complete. 
60. The relation 
C 6 C 10 C 5 C 9d~ C l C 13 C 2 C 14 = 0 
may be simplified by means of general relations between the c’s. 
By Mr. Forsyth’s product theorem, we can prove that 
^0^12^6'^ 10 ^3^15^5'^'9 ~b ^$ 8 ^ 2^14 ^7^11'^1'^13 = 
^0^12^5^9 "b ^3^15'^6'^10 - b ^-t^S'^1’^13 ^ 
@ 0 ^ 12 ^ 2 ^ lid - ^3^15'^1'^'13 ^4^8'^'6'^iO ^7^11^o"^9 = ^ 
^0^12^1“^ 13 d - ^3^15'^2"^11' @4@S^5^9 ^7^11'^6’^10“ 0. 
’ ll 
and therefore we know a, b, c,f, g, h com- 
We have already found the ratios 
/ 
