

G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 255 
We may express 2, y, z in functions of a, b, c, and supposing we have a 
function of ayz, say 7, we may express it by, or suppose it to be transformed 
into, a function of a, b,c; p, entering into the transformed function will not vary 
by differentiation. 
Calculating, as for example, <7, we put the partial differentiations under 
form: 
dr _dadr | dbdr | dedr 
da — deda* dx db + de de 
dr _dadr | 
dy = dy da + &C 
dr da dr 
dz = de da + te. 
da db 
Now = dz? &te., expressing partial differential quotients represent precisely 
the same quantities as in ay: (9), and are given by the values (10). 
: d 
If now we multiply tay iy , etc., respectively by 7,7, 4, and sum, the first 
members give <7. 
The second members give the sum 
da | .da jill 
tae td dy TAG 
as the factor of - But by the substitution of the values (10) this becomes 
— (Sar + jSay + kSak) , 
which is = + a. 
Likewise the factors of ~ , ue are found to be £, y respectively. 
Thus we have, for the expressions defined by (1), the transformations 

2a ae ON 
| Pease alt 
(10 dis) 
dr 
br =Ta+Gh+ ay 
the second of these formulas being formed in a similar way. 
This transformation shows that the operators 4, > , work out results which 
are not dependent on the particular direction of the system of treble rectangular 
axes of co-ordinates to which the variables refer, provided that in the differen- 
tiation the axes themselves remain constant in direction. 
If we differentiate the equations (9) partially in respect to any of the variables 
2, y, z, the differential quotients of a, b, c of the second order must a@// vanish. 
To this circumstance the result is due that the operators 4’, D>’, <(b), 
> (<j), as given by (6) and (7) will, like < and pb, preserve their primitive form, 
when the change of variables from 2, y, z, to a, 0, ¢, is introduced, so that we 
VOL. XXVII. PART III. 3X 
