G. PLARR ON THE ELIMINATION OF a, 8, y, ETC. 273 
49.9 = 55 iB¥| | aV(a<v) is 
which by (B), § 1, becomes =— <Jv. Likewise by second and third members 
poy =— 5 (Vadv)a =— av. 
We have thus: 
| ap.q =— dv= {KI 
(71) 4 
L 
The second member being a vector, the scalars of dp.g and of ppq are 
zero. This consequence would also follow from a comparison of the scalar of 
(56): 
peg =- dv=— 70. 
Sppqg+Sdp.g=0, 
with 
peq= Ap-q, 
giving, as corresponding to SII = 0, the conditions 
(72) Speg —0,- S4p.¢=.0: 
We treat by S<. the first of the equations (71), or by Sp. the second. 
The first gives, <’v being a scalar 
s[_<"p-¢ +2 
As we have already seen by (60), the first term is = 3p'g,; the second 
termis = Sdppq. Thus the result is 
(73) — Srp.g, + Sdppg =— <u. 
The first term by (70) is 
ijk 
LY & 


iap@ |=—%v. 
— : (an)7 
the second term will, by multiplying the equations (71) into one another, and 
owing to pg = 1, give: 
(74) Sdppq=+t Ciel), 
Thus the equation (73) gives 
— 5 (<u)? + (dv)? =— av, 
from which results (48 07s) of § 3. 
We remark, that by (70) and (74) we must have 
(75) Slipped + 23p¢5] = 0, 
| in virtue of the conditions of integrability, This equation, when developed in- 
dependently of these conditions, will not give an identity @ priori. 
