7 84 Proceedings of Royal Society of Edinburgh. [sess. 
In connection with which equations we observe the identities, 
2(B 1 C, - C ] B 5 )(E 1 E 2 - CA) = 2(E 2 C 5 E 1 B 1 + 0^ + B^EJ ; 
&c. 
In writing down the complete out-spread of W 2 , it will not be 
necessary to avail ourselves of more than a few of these results. 
18. From what proceeds we have 
W 2 = U 0 + U 1 + U 2 + U 3 + U 4 + U 5 + U 6 + U 7 + U 8 + U 9 ; 
where 
U 0 = VS 2 y8yS 2 ySS 2 S€S 2 ^S 2 Ca ' 
Uj = - 22S 2 a£S 2 a£S 2 /3yS 2 S€SySS6£SS£Sye ; 
U 2 = - 2^S 2 ay8SVS 2 S€S 2 y8SeCS/36SyCS/?y ; 
U 3 = 22S 2 a/?S 2 a£S 2 S€S£yS/?8Sy<:Sy£SySSc£ ; 
U 4 = 22S 2 a/?S 2 y8S 2 e£SaeSSeS/?SS£ySy£Sa£ ; 
U 5 = - 4^S 2 ay8S 2 y3S 2 e^Sa^S3CSaSS^ySy€S^€ ; 
U 6 = 22S 2 a£S 2 yeS 2 3£SaeSS € S£SS/?ySy£Sa£ ; 
U r = 45SVS 2 y3SacSa^S/3 € S^Sy€Sy^S8 € S3^ ; 
Ug= - 2^S 2 a/3S 2 ySSaeSa£Sy€Sy£S/3eSS£S/?SSe£ ; 
U 9 = 4SSaiSo8SaySaj8S^S«J SycSSf^SfS^cSySSSe . 
In this expansion for W 2 it is to be observed that 
U 9 contains 15 terms, U 0 , U 5 , U 6 , each contain 60 terms, 
U 7 contains 45 terms, U 1 , U 4 each contain 180 terms, 
U 2 contains 90 terms, and U 3 , U 8 360 terms each. 
Counting the weight of each term as 1, 2, or 4, according to its 
coefficient, we have, 
weight of + terms = 60 + 2 x 360 + 2 x 180 + 2x 60 + 4x45 + 4x15 
= 60(1 + 12 + 6 + 2 + 3 + 1) = 1500 ; 
weight of - terms = 2xl80 + 2x 90 + 4x 60 + 2x 360 
= 60(6 + 3 + 4 + 12) = 1500. 
The same number, as should manifestly be the case. 
19. If we had expressed W 2 in terms of b l9 c l9 e lf &c., it should 
have been borne in mind that these Scalars are connected by five 
