7 26 
MR. A. McAULAY ON THE MATHEMATICAL 
In the equation for SI put = - SvgS ^~ l <r and S^VT = - S^S^VT'; for 
8 m substitute from eq. (9); and for l”, r VT and a V'l", substitute in terms of l, T Vl and 
a Vl. Thus 
- S 8 VlQll = - S 8 V£{mQn + i (l + SSr T Vt)^-^}, 
- - SrSty-yi}. 
Now, let 12, u be two functions of Class I. of § 9, the first given by 
n = — 2(1/ + 2mQl" + (l + 2St t VZ)¥ -1 .(11), 
from which [eq. (9) § 9, ecp (32) § 50, and Prop. II. § 8] 
n' = f + 2 X ar x + v + tSr'yr .( 12 ). 
Let v be given by 
va> = "% (tS 7 V/i// _ ~ a) — a .V/Scn// - 'cu).(13), 
from which are easily deduced 
?/ft> = 2 (r'Syi'co—yi'So-'aj).(14), 
v"oj = 2 -V (qc^" 1 ) q = x (t'UVT'oi - yTSa'to ). . . . ( 15 ). 
From the last value for SZ we now have 
S = 2S Si/>£ui//£, 
or 
Hence (former paper, p. 105) the pure part of fhfj = ditto vi/j, fie., 
fhfj = vxp -j- mV 7 ] ( ), 
where rj is a vector to be determined. Hence 
CIoj = voj -f- mVrjxjj~^co .... 
Therefore 
Cl'a> = -|- x^VQ 
L ct)q 
(16). 
(17). 
and 
TCgj = i/Tj ~p xpVyjoj .( 18 )* 
I rom the last equation and the fact that £ = 0, it is easy to deduce that 
V = (* + SW)" 1 SV (cr'VV'T' + t" t V"1") 
( 19 ). 
