254 G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 
Let us represent p by the vector sum (1st) of OM, represented by p,, whose 
origin is O, and its extremity M,, M, remaining invariable when M varies by 
differentiation, and (2d) of M, M, 
represented by , so that when 
M varies w varies also, but only by 
Mo its extremity M and not by its 
origm M,. We have thus 
Pi — "Po tee. 
Through M, we conceive a system 
of treble rectangular axes, invari- 
able in direction, and we designate 
g by a, B, y, the unit-vectors corre- 
sponding to their directions. Let a, b, c, be the Carthesian co-ordinates of M 
in respect to these axes. Then we have 
w= aa+ bh+ye. 
For any displacement of M we have 
dp = do , 
namely : 
idx + jdy + kdz = ada + Bdb + ydc, 
but we consider three particular displacements corresponding to’ 
Ist; dy = 0, dz =0, dz not zero, 
od, d= 0 dr =O ey ” 
od, 4 =”. dy=05 02 
This gives us severally 
_ de, gt , ode 
=aq, + B dx * ¥ dx 
2. 
: _ da - db _ ae 
(9) j = tag PB agit hay 
_ da - db _ de 
k= aa, He as age 
The nine quotients which correspond to as many partial differentials of a, 6, ¢, 
are easily determined by the help of the properties of a, B, y, as a treble rect- 
angular system of unit vectors. Thus we get: 

da aa Oe CLG a 
Fi ee Saz, day a Sf: , ie a Syz 
da Jae cho SCLC oe 
(0) | =~ Sy, 5 =— SB, G =— Sit 
da db de 
~ =— Sak, F =— Sbh, & = — Syh. 
: 
b 

