429 
and as the perpendicular from the origin upon the tangent 
plane is equal to 
(x + y i zs 
we find the volume of the small pyramid, which has the origin 
for its vertex and dS for its base, to be equal to 
dydz 
x 
1 
3 
But again, as 
dx = zdp + ydy, 
dy = xdp + zdy, 
dz = ydp + ady, 
the element dydz must be replaced by (2* - yz) dpdy under 
the double sign of integration. Hence 
alee = 3 {J dpa. 
Thus it appears that the sector, generated by a radius vector 
drawn from the origin, and having for its base the portion of 
the surface bounded by the lines ¢=0, x=0, $= X=X’3 
may be represented by the product 3 ¢’y’- 
«< Tt has been observed that the functions obtained by di- 
viding the two tresines by the cotresine possess properties 
analogous to those of the trigonometric tangent. A remark- 
able instance to this effect may be adduced in connexion with 
the preceding theorem. 
“* Let us put 
; : z 
trit (¢, x) -2 =n, and trit (x, ¢) = a es 
then 
Dy ns 
we ady ae, aah _ dz zdx 
x 
or os es 
d (a? — yz) dp - (y?- 22) dx _ xdp - zdy 
Res Seg ee 
£ it 
and 
