268 G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 

Now we substitute these values of aw, etc. into the expression 0,. We may 
represent a, by 

= (2V7bv), 
where 7 is to represent a, 8, y, when ¢ represents a, b, c, respectively. Thus 
we have (nine expressions in one) by 
6; = V[0V(4v7)). 
Writing this for ¢ = a, b,c, and multiplying respectively by da, db, dc, and 
remembering 
dp = ada + Bdb + yde, 
we get the results-— 
(52) dB = V[BV(<vudp) | 
dy = V[y(V(<vudp)]. 
These are the differential equations which connect a, 8, y and their differentials 
with w, in virtue of the conditions of integrability. 
The consequence which follows from these equations, at first sight, is that 
when V(<vdp) = 0, the increments da, dB, dy are also zero. 
Now we have V(<vdp) = 0, sata the displacement dp is, in its direc 
tion, normal to the surface whose equation is « = constant; and therefore the 
values of a, 8, y remain unchanged along the direction of that normal, when 
the displacement is infinitely small of the first order. 
It will be interesting to see in what way the formulas (52) work out this 
result. For this end we assimilate in them a, £, y, with a, 8, y; then for the 
three components of dp, namely, the three partial displacements ada, db, 
ydc , which compose the total displacement dp, the corresponding variations 
of, as, for example, a, namely, aida, ajdb, a,de, which compose da, will be 
da = V[aV(<vdp) | 
a, 40 Y 

a,da = (Bu, + yv,)da 
(52 bes) a,db = — Bu,db B 
a.0G == — yu,de 
ie a 

namely, all three perpendicular to a, as a matter Ja),de 
of course, but the two latter ones will be of a 
similar direction to that of the components of dp to which they correspond; 
whereas the first, corresponding to the component of dp in the direction of a 
itself, will be generally oblique to the two other variations. By their sum 
> 
da = ada + adb + aide, 

