302
ON CURVATURE AND ORTHOGONAL SURFACES.
[519
and hence
d x a+d y ß + d z y =
a + b + c 8V
V V*’
Xd x « + Yd x ß + Zd xJ = y - ~ SX, = 0,
ad x p + ßd y p + y d z p = y 8p,
aX + ßY+yZ = V ;
and we have
X' = X jl + P (t
ci + b + c 8V'
. V V\
;) + y ~ Vd X P>
with the like values of Y' and Z'. But we are only concerned with the ratios
X' : V : Z’\ whence, dividing the foregoing values by the coefficient in { }, and
taking the second terms only to the first order in p, we have simply
X', Y\ Z' = X-Vd xP> Y — Vdyp, Z-Vd zP .
24. We may investigate the condition in order that the surface x', y', z may be
the consecutive surface r + dr = r (x, y, z). This will be the case of
X\ Y', Z' = X + ^8X, Y + ^8Y, Z + y 8Z,
which is as it should be, viz. these are what X, Y, Z become on substituting therein
for x, y, z the values x + poc, y-\- p/3, z + py.
25. I return to the case where p is arbitrary, and I investigate the values of
a, b, ... for the point P' on the vicinal surface; say these are a', b', &c., then we
have a' = d x X' &c. The relation between the differentials may be written
dx = (1 — d x poL) dx — d y pa dy' — d z pz dz,
dy= — d x pß dx + (1 — d y pß) dy' — d z pß dz,
dz = — d x py dx — dypy dy' + (1 — d z py) dz,