43. Singular points of a curve whose equation is, u = ƒ (xy) = 0.

d2y

changes sign for a small increase and decrease of r.

dx2

DEFINITIONS AND GENERAL NOTATION.

Indefinite Integral. fudz-That function of

becomes u when differentiated with respect to x.

Judx = f(x) + C. C'being constant.

Definite Integral. S

a

x which

u.dx-The value of the indefinite inte

gral Juda between the limits a = a, and a = b.

b

x x

f'udx = ƒ (b) — ƒ (a).

dx

Jwdx.

2. f(±±±w) fede± fuds. (± u ± v ± w) dx = ± fudx ± Juda ± Svdx

23.

24.

25.

= log(x+√x2± a2).

a2x2

a2

dx

x2

=

1

dx
a2

(a + √ a2 ± x2

1log
log (a

a

a + x

log

2a

a

Xa

2a x +

√ √ x2 ± a2. dx = 2 √ x2 ± aa ± a2 log (x + √x2± a2).

I. A Plane. (1) A surface in wl ich if any point be taken, it is equidistant from two given points. (2) A surface generated by a straight line moving always parallel to itself, and intersecting a given straight line.

II. Co-ordinate Planes.-Three fixed planes passing through the same fixed po nt called the origin of Co-ordinates, and intersecting one another two and two in straight lines called the Axes of Co-ordinates.

III. Projections of a Point.-The feet of the perpendiculars dropped from a proposed point upon the co-ordinate planes.

IV. Traces of a Plane.-The lines in which a plane intersect the co-ordinate planes.

V. Projection of a Line upon a Plane.-The locus of the projections of every point in the line.

VI. Trace of a Line upon a Plane.-The point in which the line meets the plane.

VII. Projecting Plane.-The plane which contains a proposed line and its projection.