37] 
ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 
247 
if Po, q 0 > r 0 , V 0 refer to the value v 0 of v, i.e. if 
A p 0 2 + B q 0 2 + G r 0 2 = h 
A 2 p 0 2 + B 2 q 2 + CV 0 2 = k\ 
Ap 0 a + Bq 0 b + Cr 0 c = 2v 0 - k 2 . 
{This implies evidently 
b % ~ c % = V ( hCr ~ cBq) ~ V 0 ( bCV ° ~ cBq °^ 
an equation which it is interesting to verify. In fact, from the value of e 
•(48), 
, de de 
de C db 
or we have to show that 
d 1 
d 
d\ 1 
de db V 
-2 dv b-f-c^ ~ = 2 dvA b7-c 
dV dV 
if for shortness 
dv V (bCV cBq) ~ V 2 ( b de ~ 
<v , d d 
h ~ h de~ C dV 
dV dV 
C db 
V 2 V de 
db 
Now V containing a, b, c explicitly, and also as involved in p, q, r, we have 
SV = bpq (A — B) —crp (C — A) + ^ 8p + 8q+^p~ 8r = bpq (A - B) - crp (C — A) + S'V 
suppose. The equation to be verified becomes 
V ( b6 ^ ~ cB d!u)~ ( hCr ~ cBq ^ Jv = 2 f hpq cr P (0-A) + 8'V}. 
Now, observing that 8k = 0, we have 
A p8p + B q8q + G r8r = 0, 
A 2 p8p + B 2 q8q + C 2 r8r = 0, 
A a8p + B bS</ + C c8r = — (bCr — cBq). 
dv 1 dv dv 
A^^ + Bb^ + Cc~ = 2, 
av civ civ 
Also,