566] 
SURFACE TO A SET OF CHIEF AXES. 
51 
Moreover, p lt p 2 being thus determined, we have, a lt ft, y lt Q x proportional to the 
determinants formed with the matrix 
a — , h , q , A 
Pi y 
h , b--, f , B 
Pi 
9 > f > G » 
Pi 
say, a 1} ft, 7j, ^ &33i, k(Ei, kfl, where 2h, S3j, are the determinants in 
question ; and then 1 = k 2 (2h 2 + 33i 2 + Gq 2 ), or we have 
a _ 
1_ V(2ii 2 +53 1 2 +e 1 2 )‘ 
But we find at once 
(»,...)(«!, ft, 7i)(a, ft y) = 6 1 (Aa + B/3+Cy) = 0 X V, 
that is, 
(a, ...)(a : , A, 70 (a, A = + (£■) ’ 
and in the same manner 
(a, ...) (a 2 , A, 7O (a, ft 7) = + ^2 + • 
Hence the transformed equation is 
X 2 Y 2 
VZ + i^+i- 
pl p2 
+ XZ 
Vfij 
+ YZ 
V Ho 
V№+s 2 2 +e 2 2 ) 
B, cy +&c = 0, 
where it will be recollected that V = V(M 2 + B 2 + (7 2 ). The &c. refers as before to the 
terms (X, F, ft) 3 and higher powers, which are obtained from the corresponding terms 
in 8x, 8y, 8z, by substituting for these their values 8x = a,X + a 2 Y + aZ, &c., where the 
coefficients have the values above obtained for them. It will be observed, that the 
radii of curvature are Vp 1? V p 2 , and that the process includes an investigation of the 
values of these radii of curvature similar to the ordinary one; the novelty is in the 
terms in XZ, YZ, and Z 2 . But regarding X, Y as small quantities of the first order, 
Z is of the second order, and the terms in XZ, YZ are of the third order, and that 
in Z 2 of the fourth order.