[3] Assume tan a = cos A. tan b;

-log Tan a = log Cos A + log Tan b — log r :

log Cos Blog Cos a + log Cos a-log Cos b:

then caß, under the same conditions as above.

Sixth Case: given A, B, a. To determine b:

log Sin blog Sin a + log Sin B — log Sin A.

The above remarks on

sin a sin b

sin A

may here be applied to

sin B

C, and c may be determined as above; or thus,

log Sinẞ=log Sin a + log Cos A – log Cos B :

then Caẞ, as above.

Assume tan a tan a. cos B,

log Sin Blog Sina + log Cos A - log Cos B ;

then caẞ, as above.

(W. Ch. xi; C. Ch. xviii; L. 221-62: Legendre, Trig. 84-91;

Delambre, Astron. 138-83.)

(27.) Series for the sine, cosine, &c. in terms of the arc.

(28.) Series expressing the inverse circular functions.

1 x3 1.3 25 1.3.5 x7

+

4813

33.5.7 33.52.7 35.5.7.11

15

36.59.7.11.13

3617x 37.5.7.11.13.17

&c.

B

(C. 266—302.)

+

+ &c:

2 3

2.4 5

2.4.6 7

(29.) Series for determining the value of π.

T=3,1415926535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 0938446, &c.

(Lacroix, Introd. Diff. Calc. 43-5.)

(30.) Formulæ involving impossible quantities.

sin nx=

1

(cosa±√1 sin x)TM

{(cos +/-1 sin x)" — (cos æ—√/—1 sin x)"}

· ́If (a + b√ = 1)8+n√ Cos c +/-1 Sin c,

b

=

and - = tana, then c=gx+h.log (a2 + b2),

a

log, 1=2(m −1) π-1; m being any integer.

Of these values only one is possible, which is obtained by putting m=1. (Lacr. Introd. 81.)

(31.) Formula for the sums of arcs, and multiple arcs.

Let the sum of the continued products of each combination of the sines of m of the arcs, a1, α, &c. a,, into the cosines of the remaining n-m arcs, be represented by Cn-mSm; then

sin (a1 + a2+ &c. + an) = Cn− C n − 2 S2 + Cn−4S4- &c.

cos (a1 + a2+ &c. + an) = Cn-1 S1 − Cn - 3S3 + &c.

Let T be the sum of the continued products of each

combination of m tangents,

to (n + 1) terms, if n is odd, and to in terms, if n is even.

to (n + 1) terms, if n is odd, and to n+1 terms, if n is even.