AOPI  , P20p3, p3OB and AOP3, P80p2, p3 P3 p2OB satisfy the conditions of similarity sides; thus P2, P3 represent the roots

cos js(0+2w)+t sin 1 (0+22r), cos 1 (0+41r)+t sin 3(0+42r) respectively . If B coincides with A, the problem is reduced to that of finding the three cube roots of unity . One will be represented by A and the others by the two angular points of an equilateral triangle, with A as one angular point, inscribed in the circle . The problem of determining the values of the nth roots of unity is equivalent to the geometrical problem of inscribing a regular The nth polygon of n sides in a circle . Gauss has'shown in his Roots of Disquisitions arithmeticae that this can always be done unity, by the compass and ruler only when n is a prime of the form 2n+I . The determination of the nth root of any complex number requires in addition, for its geometrical solution, the division of an angle into n equal parts . 19 . We are now in a position to factorize an expression of Factorisa- the form x"-(a+tb) . Using the values which we tions. have obtained above for (a +0)0', we have s= n-1 = 8+25>< 8+2s~r x"-(a+tb)=P [x-r" (cos n +t sin- n )]• (I) s=o If b=o, a=I, this becomes s=n—1 [ 2Sa 2Sir x"—1=P x-cos n -tsin n =(x-I)(x+I)P )n—1 (x—cos 2SIr t t sin 2n s=1 2Sa =(x— I)(x+i)P s=.n—1(x2—2x cos 2- +I) (n even) . (2) s=0 s=1(n—1) x"—I=(x-I)P (x2—2xcos2 +I) (n odd) . (3) s=1 If in (I) we put a=-1, b=o, and therefore 0=3r, we have =n—1 x"+I =P s =n-1 2- n I1r —t sin 2S+n I1r1 s=o 1 s=}(n—2) 2S+I7r =P [x2—2x cos n +1] (n even). s=o s=i(n—3) 2S+I7r x"+I = (x+i)P [x2—2x COS n +I] (n odd) . (5) s=0Also x2"-2x"y" cos ne+y2" = (xn - yn cos n6 let sin nb) (x"- y" cos nO -t sin n6) s=n-/ 0 =P ( B+2ssr t sin —=-1 s=o .x-y cos \ n n J s=n- =P [x2-2xy cos B+2n"+y2] .

(6) s=0

Airy and Adams have given proofs of this theorem which do not involve the use of the symbol t (see Comb . Phil . Trans., vol. xi) . A large number of interesting theorems may be derived from De Moivre's theorem and the factorizations which we have Example of deduced from it; we shall notice one of them . Example In equation (6) put y1/x, take logarithms, and then re Theorem . differentiate each side with respect to x, and we get 2n(x2n-1—x—2n—i) s _n.--1 2(x—x 3) x2i-2 cos n0+x-2"-ss=0 2 2Ssr x -2 cos 6+—+x n Put x22=alb, then we have the expression n (a2n - b2tt) (a2 - b2) (a'n -2 a,b, cos n0+b2") for the sum of the series s=n—1 s=o 2 2S!r a-2ab cos 0+ n +b2 20 . Denoting the complex number x+iy by z, let us consider the series 1 +z+z2/2 ! + ... +z'/n ! + . . . This series converges uniformly and absolutely for all values of z whose Then.. moduli do not exceed an arbitrarily chosen positive potlal number R .

Consequently the

function E(z), defined serienes. as the limiting sum of the above series, is continuous in every finite domain . The two series representing E(z1) and E(z2), when multiplied together give the series represented by E(zi+z2) . In accordance'with a known theorem, since the series for E(zi) E(z2) are absolutely convergent, we have E(zi) XE(z2) =E(z1+z2) . From this fundamental relation, we deduce at once that {E(z) l' =E(nz), where n is any positive integer . The number E(i), the sum of the convergent series I+I+1/2!+1/3! is usually denoted by e; its value can be shown to be 2.718281828f59 . . .. It is known to be a transcendental number, i.e. it cannot be the root of any algebraical equation with rational coefficients; this was first established by Hermite . Writing z=1, we have E(n)=e°, where n is a positive integer . If z has as a value a positive fractionp/q, we find that{E(pig) }4=E(p)=e5; hence E(pig) is the real positive value of &I Q . Again E(-p/q) XE(p/g) =E(o) =1, hence E(-p/q) is the real positive value of e p'Q . It has been thus shown that for any real and rational number x, the value of E(x) is the principal value of es . This result can be extended to irrational values of x. if we assume that e5 is for such a value of x defined as the limit of the sequence ei1, e:2 , where x2, .

. . is a sequence of rational

numbers of which x is the limit, since E(x1), E(x2) . . ., then converges to E(x) . Next consider (1+z/m)"', where m is a positive integer . We have by the binomial theorem,} r (1+m) m- i+z+ (I m/ z2 . + (I m) (I m) .. . (I—Sm1)'+..+1.m) m (I m) (I J) ... (I—'m 1 \ lies between 1 and + (m+m+ ... +s ml) ; hence the product equals i -0,s . S- I/2m where 03 is such that o<0 <i . We have now (I+m) m=i+z+ (I-m) zj+...+[I-Bes.2m 1]Si+ ... +[i-8 m-Il '' 2 = 1+z +22/2 !+ . . .

+Zs/s ! +Rs, where ,s+i z"` z2 z Rs=(s+I)!+...+nt!-zm I+B3I+ ... zs—2 zm—2 +0,(s-2) !+ . +‘(,- 2) Since the series for E(z) converges, s can be fixed so that for all values of m >s the modulus of zs+i/(s+i) ! + . . . +z'"/m! is less than an arbitrarily chosen number le . Also the modulus of I+0sZ/I +...+0n,z"-2/(m-2)! is less than that of 1+11 zI/i ! +1z 12/2 ! +..., or of emods hence mod Rs<le+(I/2m). mod (z2e') <e, if m be chosen sufficiently

great . It follows that limm_,o(i+z/m)"'=E(z), where z is any complex number . To evaluate E(z), write I +x/m = sr) cos 4>, y/m = p sin 0, then s=1 (4) Also E(z) = limn _~ {p" (cos m~ +i sin mo) }, by De Moivre's theorem .

Since p"`= (I+m) ") I+(/ m-} x/ Jm)2 , we have =es. limm ... ) I+m(Jm+x/ Jrn)2 Let r be a fixed number less than Jm+x/.,

Im, then limn,_ I+m(Jm+xJm)2'sra lies between i and limm ... ) 1+n- „ y2 ) 2 ' or between i and eve 2r2; hence since r can be taken arbitrarily large, the limit is r . The limit of m4, or m tan-'{y/(x+m)} is the same as that of my/(x+m) which is y . Hence we have shown that E(z) =es(cos y+i sin y) . 21 . Since E(x+iy) =i'(cos y+sin y, we have cos y+i sin y =E(iy), and cos y-i sin y=E(-iy) . Therefore cos y=z{E(iy) Exponential +E(-iy)}, sin y=zi{E(iy)-E(-iy)}; and using the series defined by E(iy) and E(-iy), we find that Values of cos y = I - y2!2 ! + y4 /4 ! - .. , sin y = y - y2/3 ! Trigono- + y'/5 !

- ., where y is any real number . These metrical are the v3ell-known expansions of cos Functions. y, sin y in

powers of the circular measure y . Where z is a complex number, the symbol e' may be defined to be such that its principal value is E(z); thus the principal values of e'v, e-'v are E(iy), E(-iy) . The above expressions for cos y, sin y may then be written cos y = z (e'v+e`v), sin y = Zi(e'v-e`v) . These are known as the exponential values of the cosine and sine . It can be shown that the symbol e' as defined here satisfies the usual laws of combination for exponents . 22 . The two functions cos z, sin z may be defined for all complex or real values of z by means of the equations cos y='-2(E(z)+ E(-z) }, sin z = (,,){E(z) -E(-z) }, where E(z) represents the sum-function of i+z+z2,2!+...+z"jn!+ ... For real values of z this is in accordance with the ordinary definitions, as appears from the series obtained above for cos y, sin y . The fundamental properties of cos z, sin z can be deduced from this definition . Thus cos z + i sin z = E(z), cos z - i sin z = E(- iz) ; therefore cos2z+sin2z = E (iz) . E( — iz) =1 .

Again cos (zi +z2) is given by z (E (iz, +iz2) + E (- iz, -iz2) } = z { E (izI) E (iz2) + E (- izI) E (-iz2) } or ;{EUzi) +E(-iz,)}{E(iz2)+E(-iz2)}+'-44{E(izi)-E(-iz,)} E(iz2) -E(-iz2) }, whence we have cos (11±z2) = cos z, cos 22-sin 2, sin z2 . Similarly, we find that sin (zl+z2) =sin z, cos z2+ cos z, sin z2 . Again the equation E(z) =1 has no real roots except z =o, for ev> 1, if z is real and >o . Also E(z) =1 has no complex root a+ifl, for a-ifl would then also be a root, and E(2a) _ E(a+iO)E(a-i$)=I, which is impossible unless a=o . The roots of E(z) = i are therefore purely imaginary (except z =o) ; the smallest numerically we denote by 2 ir, so that E(2i7r)=1 . We have then E(2iar)={E(2i,r)}r=1, if r is any integer; therefore 21irr is a root . It can be shown that no root lies between 2irr and 2(r+I)iir; and thus that all the roots are given by z= =2irr . Since E(y+2i,r) =E(z)E(2i1r) =E(z), we see that E(z), is periodic, of

period 2ivr . It follows that cos z, sin z are periodic, of periods 27r . The number here introduced may be identified with the ratio of the circumference to the diameter of a circle by considering the case of real values of z . 23 . Consider the binomial theorem (a+b)"=+na"-'b+n(2 i I) a"-2b2+ ..

. +n(n-I)...(n—r+I)a"-rbr+...+b”. r Putting a=eie, b=e ie, we obtain (2 cos o)"=2 cos 150+112 COS n—20 +n(n-I)2COSn-40+ 2l .. +n(n- I) ...(n- r+t)2 cos(n-2r)B+ .. . r When n is odd the last

term is 2n(n-')' ' (n+3) cos B, 1(n-I) and when n is even it is n(n-1)i .. (Zn+i) an l If we put a=eie, b= -e 'e, we obtain the formula (-1).n(2sino)"=2 cosn6-2ncos(n-2)0+n(n-i) 2cos (11—4)0 I +(—I)"-rn(nI)• r (n—r+I) 2cos(n—2r)0 ... when n is even, and (- I)1(n-i)(2 sin B)" = 2sin nB-n . 2 sin(n — 2)B+n( . 21) 2 sin(n -4)8- Iwhen n is odd . These formulae enable us to express any positive integral power of the sine or cosine in terms of sines or cosines of multiples of the argument . There are corresponding formulae when n is not a positive integer . Consider the identity log(I -px)+log(I -qx) = Expansion log(1-p+qx+pgx2) . Expand both sides of this ofSinesand equation in powers of x, and equate the coefficients of Cosines of x", we then get Multiple p"+4" = (p+q)"-n(p+q)"-2pq Arcs in Powers of +n(2, 3) (1,+4)"-4p2g2+ ... - p Sines +qx+pqx+(-I)'n(n-r-I)(n-r-2) ...

(n-2r+I) Arc. r ! (P+q)"-l'•pr4'+ .. . If we write this series in the

reverse order, we have I); 7211( +q"=2(—I);[(pq)pq) n2 1 (1,2 g) 2 +n2(n4l ?2) (pq)2 2 (±) a n2(n2— 6)(n2—42)(1,4)2 3 (P42 -q) z(1,+q)nJ when n is even, and p"+4"=2(-)n2i[n(pq)2 (1,2 q) - n(3'-I2(pq)n23 (1,2 4) 2 +n (n2 — 15 I(n2 - 32) (1,4)"4 (1,24) 6+ ... + (- )'''.21(p +q)"] when n is odd . If in these three formulae we put p=ei%,q=e-ie,we obtain the following series for cos ne: 2 cos no= (2 cos 0)"—n(2 cos o)"-2+n( 2T 3)(2 COS 0)"-4— ... +(-1)rn(n-r- I)(n-ryl2) ... -2r+I)(2 cos o)"-2r+ ... (7) when n is any positive integer; ( — I )Los uB = - 2 titcos2B+n2(n24l-22) cos4B -n2(n2—226)(ln2—42) cos8 I B + ... +(-I)22"-' cos no when n is an even positive integer; n-1 (n2- I2) (n2 — I2)(n2- 32) (-I) 2 cos nO=n cos nO- n 3l cosaB+n 5l cos B.- n-I ... +(—I) 2 2 n-1 COS "0 (9) when n is odd . If in the same three formulae we put p=ee, q= -e-0B, we obtain the following four formulae:—(-1)22 cos n0= (2 sin O)"-n(2 sin 0)"-'-+n(2 ! 3) (2 sin o)"-4- .

. . +(-1)rn(n-r I)..,.(n- 2r-i) 2 sin 0)n-2r r ( +.(a even); (Io) n-1 (- I)2 2 sinnB=the same series (n odd); (II) 12 . 2 n2(n2 — 22) 4 n2(n2 -- 22) (n2 — 42) g cos n0 = l - lsinB+ 4 • sing- ,~6 sinB + ... +2"-' sin "0 (n even); (I2) sin nO = n sin 0 - n (n23 l~ 12) sin3B + n (n'- - 15 (n2 - 32) sinSB - .. . n-1 +(—I) 2"-' sin"o (n odd) . (13) Next consider the identity 1, - q = p-q 2 I - px i - qx I - (p + 4)x + pqx Expand both sides of this equation in powers of x, and equate the coefficients of x"-', then we obtain the equation f p"-qn=(p+q)"-'-(n-2) (P+q)"-a 1,4 p-q

Analytical Definitions of Trigonometrical Functions . Expansion of Powers of Sines and Cosines in Series of Sines and Cosines of Multiple Arm +(-') 2rL (ti_ i)2n (sn-}-i) (8) +(n-3) (n-4) (p+q)n-bp2g2- ... (—I)r (n-r-I) (n-r-2) ... (n- 2r) If, as before, we write this in the reverse order, we have the series (-I)2 [n (PI i) (pq)2 1 -n(n32 -22) 1'p-1-g\ 2, )2-n-3 +(-)Tn(n-I)-(n+3) 1(n—l) ! sin B +n(n2-25)(n2-42)(1, g11(in)2~~+...+(-,)21(1, +q)"-i when it is even, and n-1 n1 n-3 (—I) 2 [(pq)2 - n 2; I'(~24)2(pq)2 + (n2—12)(n2—32) (p2 q) a(pq)n2• +...+(—I)n21(p+q)"—1J when n is odd . If we put p=eie, q=e-ce, we obtain the formulae (77 sin no= lino (2cosB)^-1 . -(n-2)(2cos B)^-'+(n - 3)2 - 4)(2 +(—I)r(n—r—I)(it-r—2 (n—2Y) (2 cos 0)"-2r-1 + .

. } (14) r where n is any positive integer; (-1)2 'sinno=sine {ncoso_n(n31 22)cos3o+n(n2-2"-5 5 ! n2-42)coss- ... +(—I)2 (2 cos o)"-' (n even); (15) (-I)n2lsin nO =sin B j I—ii22!'' cos2B--(n2-12) (2-32) cos°9- . +(—1)2 1(2 cos B)"—' (n odd) . (16) If we put in the same three formulae p=coo, q= -co, we obtain the series n-2 (-I) 2 sin no=cos0[sin"-'o-(n-2)sin"-3o+(n-3)2jn-4)sin"-SB- ... +()r(n—r—I)(n —rI—2)...(n—2r)s.nn tr-1B-I-...,(n even); (17) n-I (S- I) 2 cos no =the same series (n odd) ; (18) sin n9=cos 0) n sin 0(n2-22) sin'o+ n(n2—22) ~(n2—42) sinso+ l 3• 5• ...+(—1)2 I(2 sin o)"—' (n even); (19) cos nB=cos 0) I -n22 T-sin2o+(ns-I24(nz-32)sin'o- .. +(2 sin 0)"-' (n odd) . (20) We have thus obtained formulae for cos nO and sin nO both in ascending and in descending powers of cos 0 and sin 0 .

Vieta obtained formulae for chords of multiple arcs in powers of chords of the simple or complementary arcs equivalent to the formulae (13) and (19) above . These are contained in his work Theoremata ad angulares sectiones . Jacques Bernoulli found formulae.equivalent to (12) and (13) (Mem. de l' Academic des Sciences, 1702), and trans-formed these series into a form equivalent to (10) and (II) . Jean Bernoulli published in the Acta eruditorum for 1701, among other formulae already found by Vieta, one equivalent to (17) .

These formulae have been extended to cases in which n is fractional, negative or irrational ; see a

paper by D . F . Gregory in Camb . Math . Journ. vol. iv., in which the series for cos nO, sin nO in ascending powers of cos 8 and sin 8 are extended to the case of a fractional value of n . These series have been considered by Euler in a memoir in the Nova acta, vol. ix., by Lagrange in his Calcul des fonctions (r8o6), and by Poinsot in Recherches sur l'analyse des sections angulaires (1825) . 24 . The general definition of Napierian logarithms is that, if ex-,y=a+,b, then x+cy=log (a+cb) . Now we know that Theory of exr,Y=excos y Lex sin y; hence ex cos y=a, ex sin y =b, or ex= (a +b2) 3, y=arc tan b/a ' mir, where m Logarithms. is an integer . If b =o, then m must be even or odd according as a is positive or negative; hence loge (a+cb) =loge (a2+b2) j+ c (arc tan b/a' 2nir) or log, (a+eb) =log, (a2+b2)+ e (arc tan b/aa2n+7r), according as a is positive or negative . Thus the logarithm of any complex or real quantity is a multiple-valued function, the differ- ence ence between successive values being 27rc; in particular, T perbo the most general form of the logarithm of a real positive merry. quantity is obtained by adding positive or nega- tive multiples of 22n to the arithmetical logarithm . On this subject, see De Morgan's Trigonometry and Double Algebra, ch. iv., and a paper by Professor Cayley in vol. ii. of Proc .

London Math . Soc . 25 . We have from the definitions given in § 21, cos 0)= 3(eY+e-Y) and sin cy=3i(eY-e-Y) . The expressions, 3(eY+e-Y), i(eY-e-Y) are said to define the hyperbolic cosine and sine of y and are written cosh y, sinh y; thus cosh y=cos cy, sinh y=-c sin Ly . The functions cosh y, sinh y are connected with the rectangular hyperbola in a manner analogous to that in which the cosine andsine are connected with the circle . We may easily show from the definitions that cos2(x+0)) +sin2(x+cy) = I, cosh2 y-sinh2 y=1; cos(x+Ly) =cos x cosh y-i sin x sinh y, sin(x+cy) =sin x cosh y+c cos x sinh y, cosh(a+ll) =cosh a cosh 13+sinh a sinh sinh (a+ =sinh a cosh 13+cosh a sinh j . These formulae are the basis of a complete hyperbolic trigonometry . The connexion of these functions with the hyperbola was first pointed out by Lambert . 26. if we equate the coefficients of n on both sides of equation (13), this process requiring, however, a justification of its validity, we get Expansion when x lies between = I . By equating the coefficients of n2 on both sides of equation (12) we get g2 = sine 0+ sin' 8+22.4 sin' 8+2.4.6 sin' B+ 3 2 3.5 3 3.5 7 4 which may also be written in the form _ 2x4 24x•8 2.4.6 xe (arc sin x)2 - x2+- 3 2 +3.5 3 +3.5.7 4 +' ' ' when x is between =I . Differentiating this equation with regard to x, we get - xz=x-{-3x'+3—5xs+3.5.7x7+ ...

, if we put arc sin x=arc tan y, this equation becomes 2 2 2 . 2 2 arc tany=ly I+- I+y3.5 1+y2) + ... . (23) This equation was given with two proofs by Euler in the Nova acta for 1793• It can be shown that if mod x< 1, then for any such real or complex: value of x, a value of log, (1+x) is given by the sum of the series x' -x2/2 +x3/3 - .. We then have I log 1+x =x+~3+x5, x+ ... ; Gregory's 2 I -x 3 5 7 Series. put cy for x, the

left side then becomes log (I+0))-log (I-ry)) or c arc tan y=cnir; y3 ys y7 hence arc tan y=n7r=y-3 +5 -7+ .. The series is convergent if y lies between= ; if we suppose arc tan y restricted to values between = 47r, we have 3 6 arc tan y=y-3+5-..., (24) which is Gregory's series . Various series derived from (24) have been employed to calculate the value of 7r . At the end of the 17th century a was calculated to 72 places of decimals by Abraham Sharp, by means of the series obtained by putting arc tan y=7r/6, Series for y=1hl3 in (24) . The calculation is to be found in Calculation Sherwin's Mathematical Tables (1742) . About the same of n . time J . Machin employed the series obtained from the equation 4 arc tan J -arc tan 234 ='-i7r to calculate 2r to loo decimal places .

Long afterwards Euler employed the series obtained from 4a =arc tan 3+ arc tan 3, which, however, gives less rapidly converging series (Introd., Anal. infin. vol. i.) . T . F. de Lagny employed the formula arc tan I/-/3=7r/6 to calculate 7r to 127 places; the result was communicated to the Paris Academy in 1719 . G . Vega calculated 7r to 140 decimal places by means of the series obtained from the equation '-17r=5 arc tan 3 +2 arc tan A . The formula **vr=arc tan 2+arc tan 3+arc tan 3 was used by J . M . Z . Dase to calculate 7r to 200 decimal places . W . Rutherford used the equation 2r=4 arc tan - arc tan .'o + arc tan 61, . If in (23) we put y = 3 and 3, we have 2r=8 arc tan 3+4 arc tan +=2.4I+3 10 3.5 Toe a 2 2.4 2 +'S6 I+3 100 3.5 (too) + .

, a rapidly convergent series for in which was first given by

Hutton in Phil . Trans. for 1776, and afterwards by Euler in Nova acta for 1793 . Euler gives an ,equation deduced in the same manner from the identity x=20 arc tan ++8 arc tan A . The calculation of 7r has been carried out to 707 places of decimals: see Proc . Roy . Soc. vols. xxi. and xxii . ; also CIRCLE . o =sin o sin' B 3 sin' 0 1.3.5 - sin7 B (21) Aagle +2 3 2.4 5 +2.4.6 7 +' ) In Powers 0 must lie between the values t Zir . This equation of its Sine. may also be written in the form I x' I.3 xs I.3.5 x7 arc sinx=x+2 3+2.4 5+2 4.6 7+'•• (22) arc sin x 27 . We shall now obtain expressions for sin x and cos x as infinite products of rational factors . We have Factorize- sin x=2 sin 2sinx 2 --=23 sin-sinx don of Sine 4 and Cosine. x+2r . x+3r sin sm 4 4 proceeding continually in this way with each factor, we obtain x x-{-r x+2r x+n-Ir sin x =2"-1 sin nsin n sin n ... sin n where n is any positive integral power of 2 . Now sinx+rrsinx+n-rr x+rr rr-x r II n sin n sin n n r =sin2 -sin2n, x+ins x _ sin n =cos- .

Hence the above may be written sinx=2n-1 sine (sing -n-sin2n) (sing 91 -sin2 rx-L) ' .' ( 2kr 2x) x sin n -sin n cosh, where k = = I . Let x be indefinitely small, then we have 2"-1 r 2r kr =— I sin' rIsin2 n... sing n; hence sin2 x/n si x/n sin x =n Sinn cos n (I sin2 r/n) ( sin' 2r/n/n) sin2^k /n) ' We may write this in x =n sin x cos x (I -sin2 x/n) (I —sin, sin2 x/n sin ) R, n n sin2 r n mr/n where R denotes the product /_ sin2 x/n r/n) ( sin" x/n \J in2 x/n _ I sing m+2r/n/ (I sin2 kr/n)' and to is any fixed integer

independent of n . It is necessary, when we make n infinite, to determine the limiting value of the quantity R; then, since the limit of sin x is sin x, and that of n sin x/n cos x/n x sinx=(1-') ( x2) ( x2 ) lim R . x I 22x2 m r m= The modulus of R — I is less than (I+sin2 m+Ir/n) (I+sin2 m+2r/n) ... (I+singpkr/n) -I' where p =mod. sin x/n . Now eAP2> I +A p2, if A is positive ; hence mod . (R-I) is less than exp. p2(cosec2 m+Ir/n+ ... + cosec2 kr/n)-I, or than exp . ;ping{I/(m+I)2+ ... +I/k2}-I, or than exp . { p2n2/4m2 } - I . Now p2 =sin2 a/n.cosh2 /i/n+cos2 a/n. sinh2 fl/n, if x=a+xfl; or p2=sin2 a/n+sinh2 /3/n .

Hence limn=„ p2n2=a2+132, limn pn=mod. x . It follows that mod.,, (R - I) is between o and exp . { (mod . 4)2/ xm2 } - 1, and the latter may be made arbitrarily small by taking m large enough . It has now been shown that sin x=x(I -x2/

r2)(I -x2/22r2) (I-x2/m2r2) (I +e,,,), where mod. em decreases indefinitely as m is increased indefinitely . When m is indefinitely increased this becomes xrz xrz n= ^° x2 sin x=x (I T2 — ) (I -22,2) ... =xP ' (I -ri2,n.2) (25) This has been shown to hold for any real or complex value of x . The expression for cos x in factors may be found in a similar manner 3r-2x by means of the equation cos x=2 sin -2x 4 cos 4 , or may In the formula for sin x as an infinite product put x=lr, we then get 1=5 a 2 : 3.3 556: : :. if we stop after 2n factors in the 4.4 numerator and denominator, we obtain the approximate equation r 12 .32.52 ... 2_ I -2 22.42.62 ... (2n)2 (2nd-1) or I .3.5.:.22 n I = 11nr, where n is a large integer . This ex- pression was obtained in a quite different manner by Wallis (Arithmetica infinitorum, vol. i. of Opp.) . 28 .

We have sin (x+y)=(x+y)P(I+xn -) sinx xP(I +-I) nr or cos y+sin y cot x (I+x) \I+x+0 (I+x+2r) (1+xy2r1 ... Equating the coefficients of the first power of y on both sides we obtain the series cot x=x+x+r+x rr+x+2r +x I2r (27) From this we may deduce a corresponding series for cosec x, for, since cosec x=cot ;x-cot x, we obtain cosec xxxnrx I rrx+2r+x I2r—x+3r—x-13r+ (28) By resolving cos (x+y) into factors we should obtain in a similar cos x manner the series 2 2 2 2 2 _ 2 tanx=2x r+2x+3r-2x 3r+2x+5r-2x 5r 2x+"" (29) and thence secx=tan(-+2)—tan x=?2x+r+2 2 2x 3r2 ._2x 3r+2x+""(3°) These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating .

Glaisher has proved them by resolving the expressions for cos x/sin x and I/sin x ... as products into partial fractions (see Quart . Journ . Math., vol. xvii.) . The series for cot x may also be obtained by a continued use of the equation cot x = i {cot 2x+ cot 1(x+r) } (see a paper by Dr SchrSter in Schlomilch's Zeitschrift, vol. xiii.) . Various series for In may be derived from the series (27), (28), (29), (3o), and from the series obtained by differentiating them one or more times . For example, in the formulae (27) and (28), by putting x=r/n we get sr =It tan n) I nI II+n+I—2n_I+2n+I' r=n sin n I+nn I I—n+I 211-I 2n+l' .. If we put n=3, these become r=3113 (I-2+--5+--8+...) , 3113 I I _1+1+1 . By differentiating (27) we get cosec2 x=z2+(x+r)2 S+ (a . .r)2+(x+2,02+(x I2r)2+... putx=6,andwegetr2=9i I+52+7+Iiz+ ••• S• These series, among others, were given by Glaisher (Quart . Journ .

Math. vol. xii.) . x2 29 . We have sinh rx=rxP I+n2) , cosh rx=P (I+(2nx2+I)2) if we differentiate these formulae after taking logarithms we obtain the series Sums of and sin mr/n mr/n is unity, we have be deduced thus s sin 2X P (I n x2) = (1-4,r-) 2) () cos x=2 sin x P (1_ x I 3 rr I-5 r .. nEr2) ^=m 4x2 (26) =P O I-(2n+i)2r2) If we

change x into ix, we have the formulae for sinh x, cosh x as infinite products- sinh x=x'P o (I+) , cosh x=P (I+(2n)2r2)' Series for Cot, Cosec, Tan and Sec . Series for n derived from Series for Cot and Cosec . sin' Certain 2x coth rx- 2x2=I2+x2+22+x2+32+x2+.., Series . 2x tanh rx=l2+x2+32x2+52.x2+• .. These series were given by Kummer (in Crelle's Journ. vol. xvii.) The sum of the more general series I2n+x2n+2rz".+x2"+32n+xsn + ... , has been found by Glaisher (Prot . Lond . Math . Soc., vol. vii.) If U^ denotes the sum of the series I^'+2I—m+3 +..., V' r that of the series Im+3 +5 .., and W"' that of the series Sums of t +- - +., we obtain by taking toga- 3,,, 5 log (x cosec x) = U2 (x_) 2+2 U4 ( ) 4+ Us (x_) 6+ ... , I (3:) log (sec x) = V2 (3;1 r~ 2+2V44+3V6(.2) 6+.; and differentiating these series we get 2 Cot x =2x 2x— 4x3 — e — " • r V V4 6 2 tan x = 2222x+424x3+626x5 + ....

(32) r sr In (31) X must lie between tz- and in (32) between =y'-r . Write equation (30) in the form forTrigono- This may be also easily shown as follows . Let metrical y =cos Al x, and let y', y"... denote the

differential Functions. coefficients of y with regard to x, then by forming these we can show that 4xy"+2y'+y=0, and thence by Leibnitz's theorem we have 4xy("+2)+(4n+2)y(*+I) ~-Yt*)=o . Therefore y.= — 2 —y%y,,, ; = — 2(2n + 1)- y(„+,4xyC„+2) hence -2-NIT cot x/x=—2— 4x 4x 4x -6— —IO— -14 Replacing I x by x we have tan x= x x x z I _ 3_ ~ . Euler gave the continued fraction tan nx =n tan x (n2 — I) tan2x (n2 — 4) tan2x (n2 — 9) tan2x _ 3— 5— 7 this was published in Mem. de l'acad. de St Petersb. vol. vi . Glaisher has remarked (Mess. of Math. vols. iv.) that this may be derived by forming the differential equation (i — x2)y(" ),— (2m + 1)xy('"+I) + (n2 — m2) y(') = o, where y=cos (n arc cos x), then replacing x by cos x, and proceeding as in the former case . If we put n=o, this becomes tan x tan2x 4 tan2x 9 tan2x 1 ~- 3+ 5+ 7+ whence we have z 4 X2 z2 arc tan x = + 3+ 5+ 7.+ .'+2n+I +"' 31 . It is possible to make the investigation of the properties of the simple circular functions rest on a purely analytical basis other than purely the one indicated in § 22 . The sine of x would be Analytical defined as a function such that, if x= f dY Treatment J 011 (1 — Yz)' of Circular then y=sin x; the quantity r would be defined to Functions . 2 be the complete integral f of (Idy y2)• We should then have z —x = f t ( dy y2) Now change the variable in the integral to z, where y2+z2 =1, we then have z —x = f od (Idy z2)' and z must be defined as the cosine of x, and is thus equal to sin (1sr—x), satisfying the equation sin2 x+cost x=I . Next consider the differential equation J(dyy2)(I Z2)=0 . This is equivalent to d{Pl(1—z2)+z,l (1—y2)} =0; hence the integral is y' (I —z2) -l-z-V (I —y2) = a constant .

The constant will be equal to the value u of y when z=o; whence pi (1 —z2)+z./ (I —y2) =is . The integral may also be obtained in the form yz— '(I—y2)J(I—z2)=iI(1—u2). y z Leta=r , J o, (1—y2)' R of (dz z2) =I. m' 0 —.2); we have a+/3=y, and sin y=sin a cos /3+cos a sin 13, COs y=Cos a COS /3—sin a sin /3, the addition theorems . By means of the addition theorems and the values sin 4zr=1, cos 44Tr=o we can prove that sin (za+x)= cos x, cos (vr+x) = —sin x; and thence, by another use of the addition theorems, that sin (lr+x) = —sin x cos (sr+x) = —cos x, from which the periodicity of the functions sin x, cos x follows: We have also f 11 (dyy2)=—s loge{ s/ (I—y2)+6y}; whence loge Ix/ (I — y2) + Lyl + loge k' (1 — z2) + sz} = a constant . Therefore { (I — y2)l + Ly{ d(1 — z2)+Lzl = (I — u2) + su, since u =y when z =o ; -whence we have the equation (cos a + sin a) (cos /3 + 6 sin 13) cos (a + /3) + c sin (a + 13), from which De Moivre's theorem follows .