85 
and the equation to PD is obtained from z = f (at, y) by 
regarding at as constant; and in order that PR may touch PD, 
we must have B = at being supposed constant in the 
dy 
differentiation. Hence, substituting for A and B these values, 
the equation to the tangent plane at a point at, y, z, is 
or, as it is usually written, 
z - z = p (at' - at) + q (y - y) ; 
where p and q denote the partial differential coefficients of z 
derived from the equation to the surface, and the co-ordinates 
may be either rectangular or oblique. 
Cor. 1. The plane whose position is thus determined by 
the conditions that sections of it and of the surface, made by 
planes parallel to two of the co-ordinate planes, touch one 
another, contains, as we shall shew in the next Art., the 
tangent lines of all curves that can be drawn on the surface 
through the point in question. If the given equation to the 
surface, instead of having the above explicit form, should be 
u = F (at, y, z) = 0, then to determine p and q we have 
du du du du 
+ p = 0, — + -~.q = 0 : 
dot dz dy dz 
the differential coefficients being formed as if <3?, y, z were 
independent ; hence, substituting for p and q their values, 
the equation to the tangent plane under its most general 
form is 
Cor. 2. If y denote the angle of inclination of the tan 
gent plane to that of aty, then (Art. 29) 
1 
cos y =