256 G. PLARR ON THE ELIMINATION OF a, B, y, HIC. 
may derive the transformed expressions from (6) and (7) in replacing in them 
L,Y, 2, 4,7, k, respectively by a, b, c, a, B, y. Thus, as for example, 
d? a? ad 
(1) trap =— (+ Get Be). 
Generally the directions of a, 8, y are quite arbitrary. We may form in 
respect to them expressions like 
405 8 ay, da etc, zete. 
And when the differentiations have been effected, assimilate a, 8, y respectively 
with the directions of a, 8, y, whenever this assimilation be useful. 
In the case of a formula in which the assimilation has been made, all further 
differentiation or integration would become the source of erroneous, or at least 
uncertain, results. But the formula, nevertheless, will express a quaternion 
relation which may be useful for ascertaining the relations which exist between 
a, 8, y (and any other element entering into the formula) when considered as 
fixed data, not variable. (These formulas, on all accounts, will not contain 
such elements as = , a , etc., because a, etc., cannot be differentiated.) The 
corresponding results may be called “ approximative,” not as to actual value, 
but only as to their fitness, or better unfitness, for infinitesimal calculus. 
We will adopt the notation 
4 OE 7 da 
a,= 7, te 8 ay= daudy ? ete. 
In the use of the sign 2, whenever confusion might arise otherwise, we 
inscribe immediately after the sign 2, between bars, the letters to which per- 
mutation is to be applied, namely so: 
2|a,8,y| Fa = Fa + FB + Fy, 
BT | 4a, a) =f (a, a) + (8,8) + (9,9. 
abe 
We shall make frequent applications of the following known formulas, which 
we record here, once for all; p being any vector, we have: 
> 


(A) = |aBy| aSap a ialp 
(B) >| | aVap =— 2p, 2(Vap)a = 2p 
(CM BS) | apa =+p 
(Ds | S%pa =— p* 
Gay Spire | yagi (Oia 
> | | VapSap = 0 
> | | SaVBpVyp = + o? 
a, B, y being here any treble rectangular system of unit vectors. 


