1888 - 89 .] Rev. M. M. U. Wilkinson on Scalar Relations. 781 
amiss to show how this at once appears from the algebraic expres- 
sion (15) itself. For, if we assume 
Pl+i 1 = 
= b 5 + 
Bi 
+ ^2 > 
Pi 
-9.1- 
- B 5 + B 3 + B 4 ; 1 
1 
P-2 + 1-2 = 
+ 
vO 
O 
11 
Cx 
+ C 3 3 
P2 
-92 
= C 5 + C 2 + C 4 ; 
h • (31) 
p 3 +q 3 = 
+ 
VO 
w 
II 
Ex 
+ E 4 ; 
Ps 
= E 5 + E 2 + E 3 ; J 
have, by 
(13), 
• • (32) 
Pi+Pz + Pz = 
0; . . . . 
so that 
pi + pi +pi - 2 vlv\ - 2 vlp\ - Zplvl = 0 , 
and, 
W 2 = qt + q 4 2 + qt - 2 q\q\ - 2 q\q\ - 2 q\q* 
.... (33), 
from which terms of the form Eg, 2BiBg, &c., have disappeared. 
14. It is now evident that we may write, 
W 2 = 5o) r , s + ^ + ^; ..... (34) 
where 
= <** 5,1 + ( t ) 6>2 + <*> 5 , 3 + < 0 5>4 4 - < 0 4 ,x 
+ 0> 4 , 2 + <*>4,3 + 0> 3 ,i + < 0 3 , 2 + <*>2,l ; , . . ( 35 ) 
^<*>* = oq 4 - o>2 + <*> 3 + <*> 4 + o > 5 j ( 36 ) 
So ) rtS)t = <*>5,1,2 + <*> 5 ,i, 3 + <*>5,1,4 + <1)5,34 + 0)53,4 + <*>5,2,3 
+ <**4,1,2 + <**4,1,3 + <**4,1,5 + <**4,3,5 + <**4,2,5 + <**4.2,3 
+ • . . 
+ <**1,2,3 + 01x 2,4 + 0>x,2,5 + <<>1,4,5 + <*>1,3,5 + <*>1,3,4 ; . (37 ) 
Any one of the terms in each of these equations being found, all the 
rest may he found by simple permutation in various ways. But, as 
in such process mistakes are likely to he made, a table of the values 
o>g,i, &c.; oq, &c.; and o>5,i, 2 , &c., will he given. 
15. The table may be constructed as follows, or in many other 
ways : — 
%,i - BfBl + QCl + E?Ef - 2C 1 E 1 C 6 E 5 - 2E 1 B 1 E 5 B 5 - 2B 1 C 1 B 5 C 5 
= (BjB 5 - Cfij* + (Bj + CtXB, + C 6 )[(B X + C X )(B 5 + C 5 ) - 2C 4 C 5 - 2B 1 B 5 ] 
= (BjB- - Cjty* - (Bf - C?)(B§ - Cl) = (BA - BA) 2 ; 
<o 5 = 4B X B 2 B 3 B 4 + 4C 1 C 2 C 3 C 4 + 4E 1 E 2 E S E 4 - 2C 1 0 2 E 3 E 4 - 2E 1 E 2 C 3 C 4 
- 2E 1 B 2 E 3 B 4 - 2BjE 2 B 3 E 4 - 2BjC 2 C 3 B 4 - 2C X B 2 B 3 C 4 
= 2BjB 2 B 3 B 4 + 2C 1 C 2 C 3 C 4 + 2E x E 2 E 3 E 4 : 
