
G. PLARR ON THE ELIMINATION OF a, 8B, y, ETC. 261 
bole 
((V-da) = a5 SI + 5, 
(28) (Vf) =8.5 SI + 5m, 
(Vy) = 7.5 SI + 5a. 
According to our table of definitions we have now to calculate P and Q, 
defined in (13). : 
By squaring both members of (24), we have at once— 
Wil =— 42 9’ da =— 4P, 
Therefore 
(29) P=—7V'll. 
In squaring both members in the three results (28), we get as to value: 
>(V? da) =r {S7113a? + 2SMTSas, + Fo. 
By (18), in which we identify a, 8, y with a, B, y, the second term in the 
second member becomes 2SII x SII; the first term, owing to 2a” =— 3, is 
— 3SII?. Thus: 
(30) Q =— {SII + 5 Sei. 
This result is the same with that which would have been arrived at if in 
expressions (27) we had not made the identification of a, B, y, with a, B, y. 
The management of the formulas (27) will become easier when we pass the 
first terms of the second members into the first members, and then effect the 
squaring. The first members, after reduction, will then become 
SV? da+ [SIT. 
The second members give then the sum 
1 | 
i SS a By 
Interverting the order of summation, and applying formulas (A), (D), § 1, 
and considering moreover that generally 
SaaSBa + SaBSBB + SaySBy = SaB, 
which is zero, the sum becomes 
apy! fo S'aa + 28a,0,SaaSBat . 
abe| 
> 


ee abe| w,, 


from which (30) is deduced free of restriction. 
