G. PLARR ON THE ELIMINATION OF a, £, y, ETC. 253 
(3) | Spe = Sdo 
Veo =— Veo. 
We may remark also— 
(4) \ do + Do = 28da 
deo — Pw = 2V4o. 
In the case of a quaternion 7 = w + » and its conjugate Kr = w — w, we 
have, as <jw is a vector, 
Sar =Sdo, Var= dwt+ Veo 
Spr =Sdo, Ver= 4aw— Veo; 
and for Kr = w — a, we change the sign of w in the preceding formula 
S<aKr =—Sdo, Vakr = aw — Vo 
SpKr =—Sdo, VeKr= dw+ Veo. 
With these elements we may easily verify the two following formulas 
(5) | Kar + bKr=0 
Kpr + dKr=0, 
of which we will make an application in our § 4. 
In the second order, both double operators <7”, 77, give the same result, 
d*, dr dr 
(6) me ie tap tae): 
The two operators > (<7) and <(b7) give also results the same for both. In 
the case of a scalar w the results are identical with <q’w or b’w. But fora 
vector or quaternion generally they give the result— 
(7) = > (ar) = 
ler cere BO he pe 
= "ied + I dydn® +” ded” 
(7) 
2 2 72 
ee aC plvee ae, 
+ 4 TedyJ + J TydyI So dedy! 

ae nies an 
+4177 k aD age + koah ‘ 
Let us introduce, into <, variables related to 2, y, z, by a linear relation, in 
the following manner :—Let O be the jized origin of p, whose expression Is 
(8) p=tat+jy + kz, 
its extremity being in M. 
