INVESTIGATION OF THE CURVATURE OF SURFACES. 35 
1°. The solution of a system of four homogeneous equations giving the determinant 
a) b,, Ci dy, 
An, be, Cay ds, 
as, bs, Ca ds == 0 
ds) by Cy) dy, 
may be written (subject to a condition) in any one of the forms. 
Tints Ure Ce KE, on) 
A, + B; a Gh D; 
x y 2 w 
== LS = = — 12 
Ay B, Cy D, (325) 
XL 
YE = ete 
D 
— —= ete. 
A, 
Where A,, As, etc., are the minors of the determinant. 
Hence 
2°. The solution of the system of four homogeneous equations giving the symmetrical determinant 
Ai 4 A3 
Ge b, by 
of which the reciprocal is also symmetrical, viz : 
AA UNE Aes atl 
A, B, B, LB 
À; B; Cs Ci = 0 
A} B, CG, D, 
may be put in the shape 
Ss ee at ed Se ete (re 
Ges Ne 

For, from equations (13) we have 
A, B, = A, Bi; B, 0; = B; C,; B, Dy = D, By; 
but as the determinant is now symmetrical, we have 
B,=A,; B,=C,; B,= D, 
Hence 
A, B, = A?; B,C; = C?; B; D, = D? 
Thus 
A, = VA, B, C2 = VB, OG; P2a=V B, D, 
