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,