SECTION IV. 
ON TANGENT PLANES AND NORMALS TO CURVE SURFACES, 
AND THEIR VOLUMES AND AREAS. 
107. To find the equation to the plane which touches 
a given curve surface at a proposed point. 
Let z =* f {x, y) he the equation to the surface, and 
x, y, z the co-ordinates of the proposed point P (fig. 35) ; 
then the equation to a plane passing through P will be 
z - z = A {x — x) + B (y' — y), 
x, y, z denoting the co-ordinates of any point in it. 
Through P draw a plane parallel to zx, cutting the 
surface in the curve PC, and the plane in the line PT; 
then these must touch one another at P, in order that the 
plane may be the tangent plane to the surface at that 
point. Now, for every point in the line PT, y = y, and 
therefore its equation deduced from the equation to the 
plane is 
z - z — A (x f - x) ; 
also the equation to the curve PC is deduced from z = /0> y) 
by making y constant in it; and in order that PT may 
touch PC, we must have A = —,y being supposed con- 
dx 
stant in the differentiation. Again, through P draw a plane 
parallel to yz, cutting the surface in the curve PD and the 
plane in the line PR; then, as before, the equation to PR is