519] 
ON CURVATURE AND ORTHOGONAL SURFACES. 
301 
The Conormal Correspondence of Vicinal Surfaces. 
Art. Nos. 22 to 35. 
22. We consider a surface U= 0 (or r = r), and at each point P thereof measure 
along the normal an infinitesimal length p, dependent on the position of the point P 
(that is, p is a function of x, y, z). We have thus a point P', the coordinates of 
which are 
x\ y', z'= x + pa, y + p/3, z + py, 
where a, /3, y are the cosine-inclinations of the normal, that is, 
«, A y = j, j, Jr, if V = *J jP + Y^+Z*; 
the locus of P' is of course a surface, say the vicinal surface, and we require to find 
the direction of the normal at P', or, what is the same thing, the differential equation 
X'dx' + Y'dy + Z'dz of the surface. We have 
dx' = (1 + d x pa) dx -1- d y pa . dy + d z pa . dz, 
dy' = d x p/3. dx + (1 + dyp/3) dy + d z p/3. dz, 
dz = d x py . dx + d y py . dy + (1 + d z py) dz, 
0 = X dx + V dy + Z dz ; 
whence, eliminating dx, dy, dz, we have between dx, dy, dz' a linear equation, the 
coefficients of which may be taken to be X', Y', Z'. Taking these only as far as the 
first power of p, we have 
X' = X (1 + dyp/3 + d z py) — Yd x p/3 — Zd x py, 
or, what is the same thing, 
X' = X (1 + d x pa + d y p/3 + d z py) — Xd x pa — Yd x p/3 — Zd x py, 
with the like expressions for Y' and Z'. I proceed to reduce these. The formula for 
X' is 
X' = À {1 + p (d x a + dy/3 + d z y) + ccd x p + ¡3d y p + yd z p] 
— p (Xd x a + Yd x /3 + Zd x y) — (aX + /3Y+ yZ) d x p. 
23. I write, for shortness, 8 = Xd x + Yd y + Zd z , whence 8X, SY, 8Z = aX + hY+gZ, 
hX + bY +fZ, gX + fY + cZ, agreeing with the former significations of 8X, 8Y, 8Z; also 
Vd x V, VdyV, Vd z V=8X, 8Y, 8Z, and V8V = X8X + Y8Y + Z8Z. It is now easy to 
form the values of 
d x a, d x ß, d x y, 
dyO., dy ß, dy 1 y, 
d z a, d z ß, d z y, 
« 
X8X 
h 
Y8X 
9 
Z8X 
viz. these are -ÿ 
ys~ ’ 
V 
ys > 
V 
ys 
h 
X8Y 
b 
Y8Y 
f 
Z8Y 
V 
ys » 
V 
ys > 
V 
V s 
9 
X8Z 
f 
Y8Z 
c 
Z8Z 
V 
y 3 » 
V 
ys » 
V 
ys ’