778 
Proceedings of Royal Society of Edinburgh. [sess. 
Again, 
Say/3£Sa8/?€ - Say/?eSa8/?£ = 
(SayS/?£ + Sa£S/3y - Sa£Sy£)(SaSS/?e + SaeS/58 - Sa/?SSe) 
- (SayS/5e + Sa € S/?y - Sa0Syc)(Sa8S$; + Sa£S/?8 - Sa/?SS£) = 
sv 
Syc , S8c 
- Sa/?Sa£ 
S/?y , 
S/?8 
- Sa/3Sac 
Sy£ , SSi 
Sy{, &K 
Sye , 
SSe 
Sj8y, S/S8 
+ Sa/SSay 
S/3c , S8c 
+ Sa/3Sa8 1 S££ , Sy£ 
+ 
Say , SaS 
Sac , Sa£ 
Sy3£, SS£ 
! S/?e , Sy € 
S/Jy, SjSS 
8/8* , S« 
0 , 
SayS, 
Say , 
SaS 
Say3, 
0 , 
S/5y, 
S/?8 
Sac , 
SySc, 
Syc , 
S8c 
Sa£, 
s«, 
Sy£, 
SS£ 
jjence 
D 4 = Say/?£Sa8/?e - Say£eSa8/3£ ; . . . . (21) 
Equations (17) to (21), and considerations of symmetry, show that 
we have these three equations, 
a 2 /3 2 SySY£c - a 2 (S/?£S/?Sy<- - S/?eS/?8y£) - £ 2 (Sa£Sa8y€ - SacSaSy^) ' 
+ Say/?£SaS/?€ - Say/?eSa8££ = 0 ; 
a 2 /3 2 Sy € YS£ - a 2 (S/3SS/?€y£ - S££S/?ey8) - £ 2 (SaSSa C y£ - Sa£SacyS) 
+ Say/3SSae/3£ — Say/3£Sae/?8 = 0 ; j" ^ 
a 2 /3 2 Sy£V c8 - a 2 (S/?<rS/?£y8 - S£8S0Jye) - /? 2 (Sa€Sa£yS - SaSSa£yc) 
+ Say/?eSa£/?8 — Say/3SSa£/?e = 0 ; 
These three equations, if added together, give 0 = 0, as is obvious. 
Eliminating a 2 /? 2 , by virtue of the identity, 
SySSc£SyS V £e + Sy € S£SSyeYS£ + Sy£SSeSy£YeS = 0 • 
we have an equation of the form, 
F 1 a 2 + F 2 ^ 2 + F 3 = 0; (23) 
11. Equations (22) give two independent equations to find a 2 , /3 2 , 
leading to a quadratic equation. The solution of this quadratic 
