
G. PLARR ON THE ELIMINATION OF a, B, y, ETC, 267 
(49) 6, =— 5 VO, 
where the letter 6 represents successively a, 6, y; and ¢ independently of it 
represents a, 6, c. 
Then, by the conditions of integrability under the form (44), the equations 
(27) which determine o,, «,, a, in function of V <a, etc., will become, for 
SII = zero: 


i 2V(adv) = 3/287| oSaa 
(50) — 2V(B-4v) = | | @,Sa8 
| 
— 2V(yv) = 5 | | Say. 

Multiplying respectively by the three sets of multiplicators, 
Sac, SaB, Say, as first set, 
SBa, SHB, , as second set, 
Sya, etc., , as third set, 
and summing severally, we get for the first member, with the first set: 
2V[_— Z| aBy | (aSaa) dv] =2V (av) ; 
for the second set the result is 
2V(B<1v) ; 
and for the third it is 
2V (yeu). 
The second members will be, for the first set, =a,; because its factor will be 
2|aBy|SVaa =—@v =+1; 
and the factors of «,, aw, Will be zero, that of w, being 
Saa SBa + SaB SBB + Say SBy = SaB, 
which is zero, and so likewise for the factor of a, . 
Similar results are deduced from the second set and the third set of multi- 
plicators. Thus we have the equations (27) resolved into 
( @, = 2V(a<v) 
(51) ow, = 2V(B<1v) 
( w, = 2V(y<1v) ; 
and it must be remembered that these expressions are formed without any 
identification having been made between a, 8, y and a, B, y. 
VOL. XXVII. PART III. 4A 
