232 PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
plane, and (a, b , c,f, g, hgfcu, v, w) 2 = 0 for that of the chief cone, then, if the line be 
n : v : w=£ : v : £, we have 
(«, • -X%> *■ 1 > G w)=0 
for the equation of the polar plane, and thence 
u:v: w=a%+ht}+g?; : h%+bn+fl; : gZ+fi+d; 
for those of the perpendicular, or harmonic in regard to the circular cone ; also for the 
normal u, v, w=X : Y : Z ; whence, if the three lines are in piano, we have 
£ n , £ 
aZ+hn+gZ;, hg+h+fi;, gg+ft + ci; 
x , Y , Z 
as the equation of the principal cone. This is in the sequel written, for shortness, as 
£, K =o. 
h, K '• 
X, Y, Z I 
9. Consider any point P', not in general on the surface, in the neighbourhood of the 
point on the surface, say P ; then the point P' has in regard to the surface a polar plane, 
which plane, however, is dependent on the particular form of equation — viz. x', y 1 , z' being 
the coordinates of P', and U' the same function of these that U is of x, y, z, then the form 
11 = 0 of the equation of the surface gives’for P' the polar plane {ud M ,-\-vd y ,-\-ivd s ) U' = 0 ; 
and we may through P' draw hereto a perpendicular (or harmonic in regard to the cir- 
cular cone), say this is the normal line of P'. Then for points P' in the neighbourhood 
of P, when these are such that their normal lines meet the normal at P, the locus of P' 
is the before-mentioned principal cone. The analytical investigation presents no diffi- 
culty. 
10. Taking P' on the surface, the normal line of P' becomes the normal at a conse- 
cutive point P f of the surface (being now a line independent of the particular form of 
equation), and this normal meets the normal at P ; that is, we have the principal cone 
meeting the tangent plane in two lines, the principal tangents, such that at a consecutive 
point P' on either of these the normal meets the normal at P ; viz. we have the principal 
tangents as the tangents of the two curves of curvature through the point P. 
The plane through the normal and a principal tangent is termed a principal plane ; 
we have thus at the point of the surface two principal planes, forming with the tangent 
plane an orthogonal triad of planes. 
11. I proceed to further develop the theory, commencing with the following lemma : — 
Lemma. Given the line X«+Y«+Zw=0, and conic 
(a, b, c,f, g, hju, v, wf= 0, 
