1888 - 89 .] Rev. M. M. U. Wilkinson on Scalar Relations. 779 
equation will involve W. For. as appears from (9), we can form an 
equation containing a 2 , (3 2 , and W as follows. We have, 
w= 
Sa£, 
SaS , 
a 2 
m, 
S/?y, 
P 2 
SyC, 
SyS, 
Say 
SSt , 
Sye , 
S/J« 
S*f, 
SSe, 
Sa€ 
S8£ , 
Sa£, 
Sac , 
O 
or 
8/8* , 
Sjffy, 
ys 2 
Sy£ , 
Syc, 
Say 
SSe , 
Sy8 , 
S/3S 
SS£, 
SSe , 
SaS 
S4, 
Sy£ , 
S/8C 
On expanding this, the coefficient of a 2 /? 2 vanishes, and we have an 
equation which we may write, 
G x a 2 + G 2 /3 2 + G 3 = W ; ..... (24) 
Equations (23) and (24) are sufficient to determine a 2 and j3 2 in 
terms of the Primary Scalars. 
C. Expressions for F 1} F 2 , F 3 , G x , G 2 , G 3 . 
12. We have, 
Fj = - SySSc£(S/3£S£Syc _ S/? € s/38y£) 
- SycSS£(S/?SS/?cy£ - S££S/?cyS) 
- Sy£SSe(S/3cS££y 8 - S£SS/?£yc) 
= S££S/?SyeS.y£YSc 
+ S/?3S/?cy£S.y3Yc£ 
+ S£cS/3£ySS.ycV£S; . . . (25) 
F 2 = Sa£SaSycS.y£Y8c 
+ SaSSacy£S.y8Yc£ 
+ SaeSa£ySS.ycY£8 ; .... (26) 
F 3 = Say/3£SaS/?eS.y£YeS 
+ Say/3SSac/3£S.ySY £c 
+ Say/3cSa£/38S.yeV8£ • 
• • (27) 
