G. PLARR ON THE SOLUTION OF THE EQUATION Vpgp=0 ; 49 
+Q=Saa.a8.ay 
(8) —P=28aa8 . way 
0=2San’'a=>’/SaBo'y , 
w being the conjugate of a. 
Likewise as for any linear vector-function ¢ we have the relation 
m G"VO0=V¢q'00'O' , 
so also we have for the function a: 
—Qa'VI0e=Va'la'f , 
0,0, being two vectors whatever. . 
This last relation, combined with the equation (7 bis), F(a)=0, gives us 
other expressions for P,Q. Namely, writing 7(s)=0 under the form : 
—Qan =a +P, 
and forming the result 2Sa(__)a we get: 
—Q2Sana'a= 2Saw’a + PXa’. 
But by the expression of P we have : 
—P=28aVa'B . way=2Sa(—Q)aa, 
Thus the preceding equation gives : 
—P=2San’a + P2a’ , 
and as 2a?=—3, we get the result 
(9) 2P=2San%a . 
The expression =Sa/4|[a(a) |=0, namely 
0= >Saw°a+ P2Sama+Q2Sa’, 
in virtue of 2Sama=0 (8), gives us: 
(10) 3Q =2Sao'a. 
We may also form 2Saa(Faa)=0, which gives us: 
0=2Saa'a+ P2aw'a , 
in which Q disappears owing to its factor 2Sema=0. Replacing the sum 
2Saw’a by 2P we get: 
(11) 2P’= — Santa. 
These expressions (9) (10) (11) will be needed for further transformations. 
VOL. XXVIII. PART I. N 

