240 
ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 
The equations (6) may be written under the form 
2A.il + 2¡iGr — 2vBq = k (Ap + a) — 2Ap, 'j 
— 2XCr + 2/ril + 2vAp = k (Bq + b) — 2Bq , Y (10), 
2\Bq — 2pAp + 2vCl = « (Cr + c) — 2Cr , j 
whence also, multiplying by Ap, Bq, Or, and adding, • 
2il 2 = /c {k 2 + (Apa + Bqb + Crc)) — 2k 2 (11), 
or writing 
k 2 + {Ap& + Bcjb + Crc) = 2v (12), 
this becomes 
il 2 = Kv-№ (13); 
an equation, the geometrical interpretation of which has already been given. 
From the equations (10) we deduce the inverse system 
ail — bCV + cBq = 2\v — QAp, "j 
a(7r + bi2 — cAp = 2/jlv — £lBq, v (14), 
— a Bq + b Ap + cil = 2vv — fi(7r , J 
which are easily verified by multiplying by il, Cr, — Bq; or by — Cr, Cl, Ap ; or Bq, — Ap, XI: 
adding and reducing, by which means the equations (10) are re-obtained. Hence also 
if for shortness 
= ap + hq + cr, ^ 
V = aqr (B — C) + hrp (C — A) + cpq (A — B), \ 
we have, multiplying by p, q, r, and adding, 
fid? — V = 2v (Ap + fiq + vr) — ilh (10). 
To these may be added the equation 
il = aX. -f- lop -f- cv (11)» 
which follows immediately from either of the systems (10) or (14). 
We may also put the equations (10) under this other form, 
2Xil — 2yuc + 2zT> = k (Ap + a) — 2a, j 
2A.C + 2yu.il - 2va, = k (Bq + b) - 2b, ! (10 bis). 
- 2\b + 2yaa + 2vCl = k (Cr + c) — 2c, j 
It may be remarked now, that p, q, r are functions of v; since we have to deter 
mine these quantities, the three equations 
Ap 1 + Bq 2 + Cr 2 = h, j 
A 2 p 2 + B 2 q 2 + C 2 r 2 = k 2 , (18). 
Apa + Bqb + Crc = 2v — k 2 , j