7 80 Proceedings of Eoyal Society of Edinburgh. [sess. 
Gj = S.yeVS£. { S£SS.yy3Y£c + Sj0yS.80Vc{} 
- S.y8Ye£. { S/?eS.y/?Y£3 + S/?yS.e£Y8£} 
= SySSc^S^SS^eSyC + S/?yS/?£SSe - S/?8S/?£Sy€ - S/3yS/3eSS£) 
+ Sy€SS£(S/? e S/3£SyS + S/?yS/?8Se£ - S£SS/?eSy£ - S0ySj8£S8c) 
+ Sy£S8e(S£8S£{Sye + S0yS0cS8£ - S/?eS/3£SyS - S/?yS/2SS<:£) 
= Sy8S C £(SyGeS/%£ - S£{S£8ye) 
+ Sy€S8J(S0{S£ey8 - SyGSS/? € y£) 
+ Sy£SSe(S/?8S/?£y € - Sj8cSj8£y8); 
or 
F^G,; 
and, in precisely the same way, we have, 
*2 = 
(28) 
(29) 
Again, in the same way as we obtained Equations (19) and (20), we 
have, 
= SaySaS£e — SaeSaS£y ; 
= SaS£V. aYye ; 
0 , Saf , Sa8 
Say , Sy£ , SyS 
Sae , Sc£ , SSc 
0 , S/38 , S£y 
Sy 8 * , SSc , Syc 
m, SS£ , Sy£ 
&c. 
= S/?cS/?y8£-3/?£S/? y Se; 
= S/?y3Y.y8Ye£ 
&c. 
So that we have 
G 3 = SaS^Y. aYyc . S/3ySY. /?Yc£ - Sae£Y. aYyS . SygycY. /5YS^ ; (30) 
All our results show the power which Quaternion Expressions 
have of representing in a simple manner results which cannot he 
otherwise than complicated. We give such formulae for ~F lt F 2 , 
F 3 , G 3 , as naturally present themselves, without entering upon the 
question whether simpler can he found. 
D. Development of W. 
13. Consideration of symmetry show that, in the expansion (15) 
of W 2 , terms such as Bg , 2B!Bg will disappear. But it will not be 
