INVESTIGATION OF THE CURVATURE OF SURFACES. 
Leis 
i.e. J à = 2) (a a; + nb; + mc) 
aifs 
ae = 2 (n ay + bb; + Les) 
d fs - 
1 A, + lb, Ro 
FTG. (m az + lbs + € es) 
33 
In order that the equation of the section (5) be referred to its axes, we must have the 
co-efficient of Y Z zero. Combining this with the conditions of mutual perpendicularity 
of the three planes (3) and introducing the factor $, we get the following conditions : 

As Ag + V3 by + C3 Co = 0 
yg + bib + Oy p= 0 
ay A3 + b, bs + ac; = 0 
(6) 
The first three of these, if we consider a, b, ¢, as the unknown quantities, may be 
treated by the method of undetermined multipliers. Multiplying the second of them by 
—k, and the third by —h, and equating the sum of the co-eflicients of &,, b,, ¢., separately 
to zero, we find that the first three conditions may be replaced by three others, viz: 
pg ea = 
Dr ds — h 4 =0 
d fo 
LÀ —— — = 
En AO —— = 0 
TNT 
PEN" 
a C3 
(7) 
These, with the fourth, give the following system of equations : 
(a—k) a3 +n 03 + M C3 — h & —=0 
nas + (b—k) b, 4+ les—h bd, =0 
m dz + 103 + (e —k) C— he —=0 
ds + Di bs + © C3 = 0 
Eliminating a, b, c; and h, we get the determinant 
oe Nn, M, a, 
n, bees l, b, 
Mm, l, c—hk, Gr 
a, by, Cy 
or 
(8) 
GE BE ct) Bk | ( Le) a2-+(e +a) D (a+b) e2— 21, c-— 2 m 2 n a h | 
+ (be—P) a+ (ca —m) + (a b—n) ef 
+ 2 (mn— al) b, y+ 2 (n 1— bm) à a+ 2 (lm —e n) & bi = 0 
Sec. IIL, 1882. 5. 
(0) 
