Page:EB1911 - Volume 27.djvu/292

 therefore $$\qquad \frac{\tan \tfrac{1}{4}E}{\tan^2 \tfrac{1}{4}(C - E)} = \tan \tfrac{1}{2}s \tan \tfrac{1}{2}(s -c).$$

Similarly $$\quad \tan \tfrac{1}{4}E \tan^2 \tfrac{1}{4}(C - E) = \tan \tfrac{1}{2}(s - a) \tan(s-b);$$

therefore

tan § E={tan és tan § (s-a) tan § (s-b) tan § (s-c)}§ (25)

This formula was given by J. Lhuilier.

cos 1 a+b

Also sin 2C cos 2E-cos 2C sin § E= sin %C; 2

cos Ha - 11)

cos %C cos § E+sin 2C sin %E= -#qi cos éC; CO5 QC

whence, solving for cos § E, we get

I cos a cos b-l-cos c

COS %E= ~%—f (26)

4 cos ia cos 2b cos it

This formula was given by Euler (Nova acra, vol. x.). If we find sin 2E from this formula, we obtain after reduction iE= l.. .l,

Sm 2 2 cos ia cos éb cos és

a formula given by Lexell (Acta Petrop., 1782). From the equations (2I), (22), (23), (24) we obtain the following formulae for the spherical excess zsin22E =tan R cot R1 cot R2 cot R1

4(cot r1+cot r2+cot ra)

(cot r-cot r1-I-cot 12-I-cot ra) (cot r+cot r1-cot 12-}-cot r3)X (cot r+cot 11-1-cot r2-{~cot 13).

The formula (26) may be expressed geometrically Let M, N be the middle points of the sides AB, AC. Then we find cos MN I -l-cos a+cos b-l-cos c

hence cos lE=cos MN sec la.

4 cos éb cos in ' 2 2

A geometrical construction has been given for E by Gudermann (in Crelle's Journ., vi. and viii.). It has been shown by Cornelius Keogh that the volume of the parallelepiped of which the radii of the sphere pzéssing thropgh the middle points of the sides of the trian le are e es is sin 2.

16. Let ABCD be a spherical quadrilateral inscribed in a small circle; let a, b, c, d denote the sides AB, BC, CD, DA respectively, and x, y the diagonals AC, BD. It can easily be shown by joining the angular points of the quadrilateral to the pole of the circle that A+C=B+D. If we use the last expression in (23)

for the radii of the circles circumscribing the triangles BAD, BCD, we have

$$\sin A \cos \tfrac{1}{2}a \cos \tfrac{1}{2}d \text{ cosec} \tfrac{1}{2}y=\sin C \cos \tfrac{1}{2}b \cos \tfrac{1}{2}C \text{ cosec} \tfrac{1}{2}y$$;

Whence $$\frac{\sin A}{\cos \tfrac{1}{2}b \cos \tfrac{1}{2}c} = \frac{\sin C}{\cos \tfrac{1}{2}a \cos \tfrac{1}{2}d}$$

This is the proposition corresponding to the relation A+C= for a plane quadrilateral. Also we obtain in a similar manner the theorem

sin éx sin %y

sin B cos lb sin A cos 'dy

analogous to the theorem for a plane quadrilateral, that the diagonals are proportional to the sines of the angles opposite to them. Also the chords AB, BC, CD, DA are equal to 2 sin éa, 2 sin éb, 2 sin éc, 2 sin id respectively, and the plane quadrilateral formed b these chords is inscribed in the same circle as the spherical quadrilateral; hence by Ptolemy's theorem for a plane quadrilateral we obtain the analogous theorem for a spherical one

sin ix sin %y=sin éa sin és-l-sin éb sin id.

It has been shown by Remy (in Crelle’s Journ., vol. iii.) that for any quadrilateral, if z be the spherical distance between the middle points of the diagonals,

cos 11-1-cos b+cos c+cos d=4 cos éx cos éy cos iz.

This theorem is analogous to the theorem for any plane quadrilateral, that the sum of the squares of the sides is equal to the sum of the squares of the diagonals, together with twice the square on the straight line joining the middle points of the diagonals.

A theorem for a right-angled spherical triangle, analogous to the Pythagorean theorem, has been given by Gudermann (in CreIle's ]ourn, vol. xlii.)

Analytical Trigonometry.

17. Analytical trigonometry is that branch of mathematical analysis in which the analytical properties of the trigonometrical functions are investigated. These functions derive their importance in analysis from the fact that they are the simplest singly periodic functions, and are therefore adapted to the representation of undulating magnitude. The sine, cosine, secant and cosecant have the single real period 2; 'i e. each is unaltered in value by the addition of 2 to the variable. The tangent and cotangent have the period. The sine, tangent, cosecant and cotangent belong to the class of odd functions; that is, they change sign when the sign of the variable is changed. The cosine and secant are even functions, since they remain unaltered when the sign of the variable is reversed.

The theory of the trigonometrical functions is intimately connected with that of complex numbers-that is, of numbers of the form x+y(=√−1). Suppose we multiply together, by the rules of ordinary algebra, two such numbers we have $(x_1 + \iota y_1)(x_2 + \iota y_2) = (x_1x_2-y_1y_2) + \iota(x_1y_2 + x_2y_1).$ We observe that the real part and the real factor of the imaginary part of the expression on the right-hand side of this equation are similar in form to the expressions which occur in the addition formulae for the cosine and sine of the sum of two angles; in fact, if we put $$x_1 = r_1 \cos \theta_1, y_1 = r_1 \sin \theta_1, x_2 = r_2 \cos \theta_2, y_2 = r_2 sin \theta_2,$$ the above equations becomes

$r_1(\cos \theta_1+\iota \sin \theta_1) \times r_2(\cos \theta_2 + \iota \sin \theta_2) = r_1 r_2(\cos \overline{\theta_1 + \theta_2} + \iota \sin \overline{\theta_1 + \theta_2}).$

We may now, in accordance with the usual mode of representing complex numbers, give a geometrical interpretation of the meaning of this equation. Let P1 be the point whose co-ordinates referred P to rectangular axes Ox, Oy are x1, y1; then the point P1 is employed to represent the number x1+ιy1. In this mode or representation real numbers are measured along the axis of x and imaginary ones along the axis of y, additions J, being performed according to the parallelogram law. The. points lg A, A1 represent the numbers ±1, the points a, a1 the numbers ± Let P2 represent the expression x2+Ly2 and P the expression (x1+ Ly1)(x2+Ly2). The quantities r1, 01, r2, 02 are the polar coordinates of P1 and P2 respectively, referred to O as origin and Ox as initial line; the above equation shows that r1 fg and are the polar co-ordinates of P; hence OA: OP1:: OP2r OP and the angle POP2 is equal to the angle P1OA. Thus we have the following geometrical construction for the determination of the point P. On OP2, draw a triangle similar to the triangle OAP1 so that the sides OP2, OP are homologous to the sides OA, OP1, and so that the angle POP2 is positive; then the vertex P re resents the product of the numbers represented by P1, P2. lip x2-E-Ly2 were to be divided by x1+Ly1 the triangle OP'P2 would be drawn on the negative side of P2, similar to the triangle OAP1 and having the sides OP', OP2 homologous to OA, OP1, and'P' would represent the quotient.

18. If we extend the above to n complex numbers by continual repetition of a similar operation, we have—

$$(cos \theta_1 + \iota sin \theta_1)$$ (cos 02 + L sin 02)   (eos 0,1 + L sin 02) gi =cos(01= 02 +.  + e, .> + L Sin <01 +02 +    +o, ,) -

If 01=02=. .=021=01, this equation becomes (cos 0-I-L sin 0)" =cos 110-I-L sin 110; this shows that cos0 +L sin 0 is a value FIG. 8.

of (cos 110-l-L sin n0)§. If now we change 0 into 0/n, we see that cos 0/n+L sin 0/n is a value of (cos 0-l-L sin 0)Ti; raising each of these quantities to any positive integral power m, cos m0/n-l-L sin m0/n is one value of (cos 0-|~L sin 0)Lfl. Also cos (- m0/n) + L sin (-m0/n) =;

hence the expression of the left-hand side is one value of (cos 0+ L sin 0)'”'/“ We have thus De l/1oivre's theorem that cos k0-I-L sin k0 is always one value of (cos 0-I-L sin 0)'°, where k is any rational number. This theorem can be extended to the case in which k is irrational, if we postulate that a value of (cos 0+L sin 0)" denotes the limit of a sequence of corresponding values of (cos 0-l-L sin 0)", , where k1, k2. k .. is a sequence of rational numbers of which k is the limit, and further observe that as cos k0-l-L sin /20 is the limit of cos k,0+L sin k, ,0.

The principal object of De Moivre's theorem is to enable us to find all the values of an expression of the form $$(a+\iota b)^{m/n}$$, where m and n are positive integers prime to each other. lf a=r cos 0, b=r sin 0, we require the values of r"'“" (cos 0+L sin 0)"'/" One value is immediately furnished by the theorem; but we observe that since the expression cos 0+¢ sin 0 is unaltered by adding any multiple of 21r to 0, the n/mth power of r""" (cos m.0+2s1r/n+L sin m.0+2s1r/n) is a-l-Lb, if s is any integer; hence this expression is one of the values required. Suppose that for two values 51 and 52 of s the values of this Expression are the same; then we must have m 0+2s11r/n-m.0+2'§ 1T/n; a multiple of 21r, or S1-52 must be a multiple of 11. Therefore, if we give s the values o, I, 2, .n- I successively, we shall get n different values of (a+Lb)"'"', and these will be repeated if we give s other values; hence all the values of