226 Proceedings of Royal Society of Edinburgh. [sess. 
and that corresponding to these three there are other three, viz., 
clq.bnp.a.q.n , dq.bnp.m.c.p , clq.bnp.r.l.b . 
The sum of the first three we may denote by 
o 
bnp.dq.'SA.p.l , 
of the second three by 
© 
dq.bnp.^a.q.n , 
and of the whole six by 
* o 
bnp.clq.%%A.p.l 
o 
if we agree to use % in connection with the cyclical substitutions 
of § 2, and 'Z in connection with the interchange of § 8. 
(12) For the purpose of calculating the final expansion of (y) it 
is greatly advantageous to throw it into the form 
cr - pi «! ft 
ftj a ~ Pi a 2 
a 3 ft ^ ~ Pi > 
which is authorised at once by the agreement that 
a = AMR + amr 
r p 1 = A.p.l + a. n.q 
p 2 = 'M.b.q + m.p.c 
\ p s = ~R.n.c +r.b.l 
r = RA.w + mr.b - A.m.p - bn.q 
a 2 = AM.j)+ rant -M .r.b -np.c 
V a 3 = MRA + am.p -R.a.% -pb.l 
ft = ra.l + MR.c - alNL.q - d.p 
d /3 2 = am.q + RAJ - m.R.c - Iq.b 
V /-?. — m.v f 4- A M n — v A Z — qc.n 
f 
vft = mr.c +AM.q-r.A.l 
where it is observed 
(1) that cr j, p v p 2 , p 3 are symmetrical with respect to the 
interchange ; 
(2) that <*!, a 2 , a 3 are the counterparts of ft, ft, ft with 
respect to the interchange ; 
Pi a i ft 
r\„. 
(3) that 
2 3 
Pi a 3 ft 
coexistent with the cyclical substitutions of § 2. 
ft are cycles 
