Page:Encyclopædia Britannica, Ninth Edition, v. 23.djvu/588

the sphere passing through the middle points of the sides of the triangle are edges is sin - .

Properties of spherical quadrilateral inscribed in small circle. Let ABCD be a spherical quadrilateral inscribed in a small circle ; let a, b, c, d denote the sides AS, BC, CD, DA respectively, and x, y the diagonals AC, SD. 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 whence .ad ?/.& C y sin A cos x cos ~ cosec 5 = sin C cos ^ cos = cosec ^ SB * 9 ^ 82 2 sin (7 b c cos -cos g a d cos ^ cos x. This is the proposition corresponding to the relation A + C=irfor a plane quadrilateral. Also we obtain in a similar manner the theorem sin B cos - sin yf cos 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 , 2 sin |, 2 sin 1 2 sin f m m W m respectively, and the plane quadrilateral formed by 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 . x . y . a . c sm 2 Sm 2 =Sm 2 S1U 2

It has been shown by Remy (in Crete s Journ., vol. iii.) that for any quadrilateral, if z be the spherical distance between the middle points of the diagonals, cos a + cos b + cos c + cos d = 4 cos Jar cos y cos z. This theorem is analogous to the theorem for any plane quadri lateral, 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 Gudermaun (in Crelle s Journ., vol. xlii.).

Analytical Trigonometry.

Periodicity of functions. 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 ana lysis from the fact that they are the simplest singly periodic functions, and are therefore adapted to the representation of undu lating magnitude. The sine, cosine, secant, and cosecant have the single real period 2ir ; i.e., each is unaltered in value by the addi tion of 2w to the variable. The tangent and cotangent have the period ir. 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 func tions, since they remain unaltered when the sign of the variable is reversed.

Connexion with theory of complex quantities. The theory of the trigonometrical functions is intimately con nected with that of complex quantities, that is, of quantities of the form x + iy (t = V - 1 ). Suppose we multiply together, by the rules of ordinary algebra, two such quantities, we have . . 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 a^riCOUtfj, Vl = ri mn0 lt X 2 =r 2 cose 2> y^r^sinO^ the above equation becomes r^cos 0j + 1 sin X ) x r 2 (cos 2 + 1 sin 2 ) = r^cos 6 l + 2 + 1 sin 0j + 2 ). We may now, in accordance with the usual mode of representing complex quantities, give a geometrical interpretation of the meaning of this equation. Let P a be the point whose coordinates referred to rectangular axes Ox, Oy are a^, Vl ; then the point P 1 is employed to represent the quantity x 1 + iy 1 . In this mode of representation real quantities are measured along the axis of x and imaginary ones along the axis of y, additions being performed according to the parallelogram law. The points A,A 1 represent the magnitudes 1, the points a, ^ the magnitudes i. Let P 2 represent the expression x. 2 + ty. 2 and P the expression (x 1 + iy l )(x 2 + iy 2 ). The quantities ^Vfn^VAj are the polar coordinates of P and P 2 respectively referred to as origin and Ox as initial line ; the above equation shows that 7y- 2 and 0) + 0., are the polar coordinates of P ; hence A

OP l : .OP ;t :OP and the angle P0P 2 is equal to the angle

P]OA. Thus we have the following geometrical construc tion for the determination of the point P. On OP 2 draw a triangle similar to the triangle OA P l so that the sides OP 2 , OP are homologous to the sides OA, OP, and so that the angle POP 2 is positive ; then the vertex P represents the product of the expressions represented by Pj.Pa. If x 2 + iy 2 were to be divided V by x 1 + iy l , the triangle OPP Z would be drawn on the negative side of P 2 , similar to the triangle OA P l and having the sides OP, OP 2 homologous to OA, OP, and P would represent the quotient.

De Moivre's theorem. If we extend the above to n complex quantities by continual repeti tion of a similar operation, we ^ have (cos X + 1 sin 0j) (cos 2 + 1 sin 2 ) . . . (cos B + 1 siu 0,,) 2 + ... +0 n ) + 1 sin(0 1 + 2 + ... If 0j 2 ... =0 n = 0j, this equation becomes (cos + tsin 0)" = cos nO + 1 sin ?i0 ; this shows that cos + i sin0 is a value of (cos n6 + 1 f / f i sin ?i0) n. If now we change into -, we see that cos- + t siu - is a i n n n value of (cos + 1 sin 0) ; raising each of these quantities to any positive integral power m, cos_ + ts i n is one value of (cos m n n + tsin0)". Also (TO. . / m, )0 + tsinl --0 )= n) n / m . TO COS 0+i sin n hence the expression of the left-hand side is one value of

m

x , . sin 0)" 1 1 or of (cos + isin0)~n~. We have thus De Moivre s theorem that cos + 1 sin k6 is always one value of (cos + 1 sin 0)*, where k is any real quantity.

The n roots of a complex quantity. The principal object of De Moivre s theorem is to enable us to The n c j 11 ,, , , roots of a nnd all the values ot an expression of the form (a + ib) n , where m com piex and n are positive integers prime to each other. If a=rcos0, auan titv m m * " 6 = rsin0, we require the values of r" (cos + t sin 0)". One value is immediately furnished by the theorem ; but we observe that, sincn the expression cos + tsin is unaltered by adding any multiple of o . a 1 n i. e -/ m.0 + 2s7T . m.0 + 2s7rY 2irto0, the -thpowerofr"! cos htsm hs a + ib, m n n / if s is any integer ; hence this expression is one of the values re quired. Suppose that for two values s 1 and s 2 of s the values of this expression are the same ; then we must have M 1?r - m + 2 V ; n n a multiple of 2ir or 5 X - s 2 must be a multiple of n. Therefore, if we give s the values 0, 1, 2, ... n - 1 successively, we shall get n differ ent values of (a + tb) n , and these will be repeated if we give s other m values; hence all the values of (a + ib)" are obtained by giving s the values 0, 1, 2, . . . n-l in the expression r"(c< + t sin Y wherer=(a 2 +& 2 )iand = arc tan -

We now return to the geometrical representation of the complex quantities. If the points B lt B& B s> ... B n repre sent the expression * + iy, (x + a/) 2 , (x + lyf, . . . (x + iy) respectively, the triangles OAB V OB^, . . . OB n _ l B n are all similar. Let (x + iy) n = a + ib, then the con-" verse problem of finding the nth root of a + ib is equivalent to the geometrical problem of describ ing such a series of triangles that OA is the first side of the first triangle and OB n the second side of the ftth. Now it is obvious that this geometrical problem has more solutions than one, since any number of com-, plete revolutions round may be made in travelling from B l to B n . The first solution is that in which the vertical angle of each triangle

is ${\frac {1}{n}}B_{n}OA$; the second is that in which each is ${\frac {1}{n}}(B_{n}OA+2\pi )$, in

this case one complete revolution being made round; the third

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