ELEMENTARY PRINCIPLES OF QUATERNIONS. 187 
Calculating D,, D,, D,, and substituting for [(Up)? — ?]az, &c., their values 
— D,f, &c., we get, after all reductions are made, the equation 
aS 
Tp Ta sin po 
O=Tt x ipa ss a a ee a 
—a2/'?+—yJ/PO+0—2/04+80 
Now, Tp To sin a when expressed by 
YOpp e+e +2) — arly + we, 
is an irrational function of 2, y, z; it cannot, therefore, be equal to a rational 
function, whatever be 2, y, z. Therefore the coefficient of f cannot be equated 
to zero without giving rise to an equation between a, y, z, which must not be. 
Therefore we must have f = 0. 
Let us designate by } the common value of the square of a versor. Then 
Uy = 
This value can be but a numerical value, because it must represent the value 
of (Up)’ in any direction of p. Therefore we agree to (15). 
b = numerical constant. 
This conclusion is an immediate consequence of the similar admission in respect 
to g, and it bridges over the gulf which we set a priori between a vector and 
a scalar, as the square of a versor (and therefore of a vector) is to be now 
a scalar. 
We have now established identity between ¢, and ¢,, and between ¢, and 
¢,. Namely, we have now 
@, = Tp Ta [h cos pa + g Ur sin po] 
“A RAN 
a= Tp To Lb COS pw — QJ Ur sin pa; 
and introducing in these expressions the values 
WN. 
Tp To cos pw = ax + by + ez, 
(lip Wey sin pa) Uz = 7 (02 — cy), 
+ j (cx — az), 
+k (ay — be), 
we transform by this the expressions of $,, ¢, into ¢,, ¢, respectively ; and this 
identity is due to the relations agreed to, which give 
VOL XXVII. PART II. 3 C 


