13. If log z = (ay + bx) + ↓ (ay — bx), then

18. Eliminate the arbitrary functions from

z=x8 (ax+by)+y¥ (ax+by).

22. Eliminate the functions from y = xƒ (z) + $ (z).

Result. The same as in Ex. 16

23. If z+mx+ny = ƒ {(x − a)2 + (y − b)2 + (z −c)2}, then

dz

dz

{y—b―n (z−c)} dx − {x—a—m (z−c)} dy = n(x− a)—m (y—b)

24. If z = x2 (ax + by) + $ (y2 + x2) + ¥ (y2 − x2),

then

--

25. If z={x+f(y)}, then

+

=

y*

27. If u+y+z = x2ƒ{x (u − y), x (y — 2)}, then

du

du

+(u + z)

dx

dy

du

+(u+y) √z2 = y + z.
dz

28. If u=4{F(y2 — xz), f (3zy 2y3

29. If u = xyz. F{ƒ1 (x2 + y2+ z2), ƒ,(xy+xz+yz)}, then

32. Shew how to eliminate the n arbitrary functions from

CHAPTER XVIII.

TANGENT AND NORMAL TO A PLANE CURVE.

257. DEF. Let P, Q, be two points on a curve, and suppose a straight line drawn through them; the limiting position of this line, as Q moves along the curve and approaches indefinitely near to P, is called the tangent to the curve at the point P.

To find the equation to the tangent at a given point of

a curve.

Let x, y, be the co-ordinates of the given point P,

x+Ax, y + Ay, the co-ordinates of another point Q on the

curve.

Then x', y', being current co-ordinates, we have for the equation to the line PQ,

258. DEF. The normal to a curve at any point is a straight line drawn through that point perpendicular to the tangent at that point.

To find the equation to the normal at any point of a curve. Since the equation to the tangent at the point (x, y) is

the equation to the normal at the same point is

supposing the axes rectangular.

259. Let the tangent and normal at the point P meet the axis of x at the points T and G respectively; draw the ordinate PM; then