G, PLARR ON THE ELIMINATION OF a, B, y, ETC. 257 
§ 2. The Quaternion Functions of a, 2, y. 
By definition we designate by I, II, III, IV, P, Q, R, I’, IT’, the following 
expressions— 
(12) \ f= aa. Til = > ta.a 
I = Sapa; ii =i a.0 
P= >Siaa , 
(13) R= 2VdaSda 
QO = 2V* 4a, 
(14) i =Sa7a. Ti” = Dap(<a) . 
The four first of these sums are expressible by one of them. We choose II, 
which we will write here (also for other purposes) explicitly : 
(hE = a(o;a,, = ab + a:y) 
(15) + B(Baat+ BiB + Bey) 
+ y(yaa + YB + y7) 
The terms which compose I are conjugates, with reversed sign, term by term, 
to the terms which compose IV. Example—In I we have aaa,, the cor- 
responding term in IV. is 
a,aa =— Saaa, + Vaaa,, 
and likewise for all the nine terms of [and IV. Therefore 
IV =— KI, 
The terms of III are conjugates of those of II, with reversed signs. Example— 
In III we have aaja =— Saaja + Vaaza, and so on, 
Therefore 
III =— KII. 
The relation between I. and II. may be derived from the remark that 
Sda=Spa, Vda=—Vpa. 
Thus we get: 
I = YaS<da + SaVda 
IT = YaS da — 2aV da. 
By addition and subtraction we deduce : 
TI = 23aS<da —Il 
(16) | 
I =I + 25aV da. 
