PREFACE.

Ir has been found, almost invariably, that students

beginning the Calculus meet, at the outset, with a

stumbling-block.

The Differential Co-efficient is

shrouded in a haze. The few pages which follow

may help to bring the idea of a Differential Co

efficient more within the grasp of beginners.

5th June, 1884.

ALEXANDER KNOX.

V. Function, differential co-efficient, differen-

tial co-efficient of a simple function,

Differential co-efficient of x2,

25

VI.

28

VII. A falling body, .

32

VIII. Differential co-efficient of 1o, 2o, 3o and 42, .

35

IX. Differential co-efficient of 13, 23, 53 and 71,

and the second and third differential

co-efficients, .

40

XI. Differential co-efficient of an inverse function,

46

XII. Newton's Lemmas VI. and VII.,

51

XIII. Differential co-efficients of the Trigonomet-

rical functions (geometrically),

XIV. Differential co-efficients of the inverse

Trigonometrical functions,

XV. The value of x, .

XVI.

Differential co efficients of the Trigonomet-

rical functions (arithmetically),

XVII. Differential co-efficient of a logarithmic

Maclaurin’s

ON CERTAIN INFINITESIMALS, LIMITS,

AND

DIFFERENTIAL CO-EFFICIENTS.

I. Point, Line, and Superficies

1. A point is defined as "that which has no parts and no magnitude." In order to obtain some more precise comprehension of the meaning of this term point, the following considerations may be of assistance

2. If we take two circles, having the same centre, and take any point in the circumference of the outer

B

circle, and join this point with the centre by drawing a straight line between the two points, it is evident that