48 
[566 
566. 
ON THE TRANSFORMATION OF THE EQUATION OF A SURFACE 
TO A SET OF CHIEF AXES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873), 
pp. 34—38.] 
We have at any point P of a surface a set of chief axes (PX, PY, PZ), viz. 
these are, say the axis of Z in the direction of the normal, and those of X, Y in 
the directions of the tangents to the two curves of curvature respectively. It may 
be required to transform the equation of the surface to the axes in question; to 
show how to effect this, take (x, y, z) for the original (rectangular) coordinates of the 
point P, x + Sx, y+ Sy, z + Sz for the like coordinates of any other point on the 
surface, so that (Sx, Sy, Sz) are the coordinates of the point referred to the origin P ; 
the equation of the surface, writing down only the terms of the first and second 
orders in the coordinates Sx, Sy, Sz, is 
ASx + BSy + CSz + \ (a, h, c, f, g, h) (Sx, Sy, Sz) 2 + &c. = 0, 
where (A, B, G) are the first derived functions and (a, h, c, f g, h) the second derived 
functions of U for the values (x, y, z) which belong to the given point P, if U = 0 
is the equation of the surface in terms of the original coordinates (x, y, z); we have 
X, F, Z linear functions of (Sx, Sy, Sz); say 
Sx 
Sy 
Sz 
X 
a i 
A 
7i 
Y 
a 2 
A 
72 
Z 
a 
/3 
y 
that is, X = oQSx + fi^y + y^, &c. and Sx = cqX + cl 2 Y + clZ, &c. where the coefficients 
satisfy the ordinary relations in the case of transformation between two sets of rect 
angular axes; and the transformed equation is therefore 
A (a x X + cxoF-f aZ) + B (&X + &F+ $Z) + C(y,X + y 2 F+ yZ) 
+ (a, h, c, f g, h) (a t X + a 2 Y+ aZ, &X + 0 % Y+¡3Z, ^X + y 2 F+ yZf = 0,