55

or c (sec. <p — 1 — x) =. a.

a

Sec. <p — 1 — x

And

c m log. tang. (45° + J <p) = b — - cyc \in log. tang. (45° + £ <p) + «/] = bb

Q = ----- ,

m log. tang. (45° + £ <p) 4- y

Consequently

a b

Sec. <p — 1 — x ’ m log. tang. (45° £ <p) + y

Or

m log tang. (45° + £ <p) + y

an equation,

sec. q> — 1 — x

in which <p is the only unknown quantity;and hence the value of cp may be easilyfound by approximation.

It may be further remarked, on thisequation, that the value of m log. tang.(45° + -J- (p) is the number in column y, inthe table opposite the required z. tp, andthe value of sec.p— 1 is the number incolumn x, opposite the same angle.

We may likewise transform the foregoingequation into the following, viz.

Log. tang. (45° + *■$>) + | b

sec. <p ■

- 1 — xE 4'

am