266 G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 
We have by (45) 
(47) P =—;V*ll =— 4(4v)’. 
Then by (44) being substituted for V<da, etc. in Q, we have 
LV’? da, namely Q = 2/aPy| V(adv), 
which by E, § 1, gives 
(47 bis) Q =2(40)** 
Thus we get 
(48) Slt =— 24v)* . 
Equating this value to that of (46) we have 
(48 bzs) 4<q’v + 2(<v)? = 0, or <4?(wi) = 0. 
In this equation (48 bcs) we may reintroduce the original independent 
variables, z, y, z, and the corresponding unit vectors, 7,7, 4; the change of 
variables will, according to what has been established in § 1, require no other 
transformation than putting p) = 0, and substituting 2, y, z, 2,7, k, respectively 
tOud, 6, 0a, Bye 
The remark may be made that the function I, = 2a<ja, whose vector is 
zero a priort has also its scalar, =— SII, annulled by the conditions of in- 
tegrability, so that I, as well as IV, have to vanish altogether. 
We may also notice that it is through the conditions of integrability being 
fulfilled that a connection between P = 2S’<Ja, and Q = 2V’ dais established, 
the former being @ priori = — 7V0 , and the latter becoming by the conditions 
= 2(<1v)’, which by the consequence of (44), namely, by (45), becomes 
= aVeIl. So that P and Q satisfy a posteriori to the relation P + 2Q =0. 
When the expression of wu has been derived from (48 67s), and consequently, | 
by (42), vand <u are known expressions, then the differential equations of 
a, 8, y, may be formed in the following manner. 
Supposing 
da = a,da + adb + acdc, 
which expression may be replaced in the result by 
da da da 
mn at dy i 7 
we have the expressions (21) of aj, etc., which may be comprised in the 
formula 

