572 TRIGONOMETRY [ANALYTICAL. or 2 " 7 % = V7i7r, where n is a large integer. This ex- 1 . O 9 * 6*ft> * pression was obtained in a quite different manner by Wallis (Arith- metica infinitorum, vol. i. of Opp.). Series for We have ^ + ^ p f 1 + cot,cosec, sin (x + y) __ V tan, and sinx sec. or cos y + sin y cot x Equating the coefficients of the first power of y on both sides we obtain the series = x + xT^r x^rx + 27r x-2,r From this we may deduce a corresponding series for cosec x, for, since cosec x = cot~ -cot x, we obtain 1111 11 1 cosec x = x x + ir x-ir x+2ir x-2ir x + 3ir x-3ir By resolving ^ z) into factors we should obtain in a similar cos a; manner the series 22222 tanx= -- ^ r-^- + . "ir-2a; and thence 37r-2a: +...(29), .. ....(30). These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating. COS X Glaisher has proved them by resolving the expressions for and . ... as products into partial fractions (see Quart. Journ. smx Math., vol. xvii.). The series for cot a; may also be obtained by a continued use of the equation cotx = ^l cot ^ + cot g J (see a paper by Dr Schrbter in Schlbmilch s Zeitschrift, vol. xiii. ). Series for Various series for IT may be derived from the series (27), (28), (29), TT derived (30), and from the series obtained by differentiating them one or more from times. For example, in the formula (27) and (28), by putting series for ^T t cot and n cosec. irf, 1 1 1 1 TT = n tan -11 -- ? + ; - - ;- + n = ... r , n I Ti-1 + l 2n-l 271+1 ; . 7T/, 1 1 1 1 1 m 7U 1+ 77^1~7 l +T~27i-l + 277+T i> J ; if we put n = 3, these become i_i_i i i 2 4 5 + 7 + 8"V By differentiating (27) we get coseca;=- z=, and we get ir z = (x- irf (x + 2 (x-2ir) 2 These series, among others, were given by Glaisher (Quart. Journ. Math., vol. xii.). Sums of We certain series. we differentiate these formulae after taking logarithms, we obtain the series 1 1 1 , . .. (a: 2 / y? 1 +>). cosh iry= P[ 1 + I ; if n*/ 2n + l 2 / . rt, 2a: CO 2a^~ gtanh ^= These series were given by Kummer (in Crelle s Journ., vol. xvii.). The sum of the more general series + . . . , has been found by Glaisher (Proc. Land, Math. Soc., vol. vii.). Certain If in the series (12) and (13) we put n = 2x, 0=^, we get series for sine and cosine. S T~ ""| x* a?(x>-l*) 15 >...}. These series were given by Schellbach (in Crelle s Journ., vol. xlviii. ). TT 2a/ If in the same series (12), (13) we put = o> n= we get cosx=l- . - 4V 2 ) 4x 2 (4a! 2 -2V 2 ) 1.2.3.4JT* 1.2.3.4.5.67T 6 ! -7r 2 )(4x 2 -3V 2 ) + ..., Sin X- - - _ _ 5- -r 1 o Q 4 c^ 7T J. A IT lJbt) 1 Vva We have of course assumed the legitimacy of the substitutions made. These last series have been discussed by M. David (Bull. Soc. Math, de France, vol. xi.) and Glaisher (Mess, of Math., vol. vii.). If U m denotes the sum of the series =r^,+^+ 5+ > V m that Sums of powers of the series ^ + q^+ ?^+ - > an( i W m that of the series - ^ of reci " + = sr-+ ..., we obtain by taking logarithms in the formulie . u ~ 5" 7 m ral uum- (25) and (26) log (x cosec x) = Lers. and differentiating these series we get (32). In (31) x must lie between TT and in (32) between ir. Write equation (30) in the form -,. ,. (27l+l)7T secx=S(-l) n - and expand each term of this series in powers ,of a; 2 , then we get secx= -- -j ? I / h (33), where x must lie between JTT. By comparing the series (31), (32), (33) with the expansions of cot x, tan x, sec x obtained otherwise, we can calculate the values of Uz, U 4 . . . Vz, V t .. . and W v W z When U n has been found, V n may be obtained from the formula For Lord Brounker s series of TT, see SQUARING THE CIRCLE (vol. Cou- xxii. p. 435). It can be got at once by putting = 1, b=3, tinued & 2 factors . . ,, , , ,, 111 1 0=5. ... in Liners theorem = rH --- .... = i -- 1 .... t a b c a+ b-a+ c-b + for TT. Sylvester gave (Phil. Mag., 1869) the continued fraction TT_ J_ 1.22.33.4 2~ + 1+ 1+ 1+ 1+ " which is equivalent to Wallis s formula for IT. This fraction was originally given by Euler (Comm. Acad. Petropol. , vol. xi. ) ; it is also given by Stern (in Crelle s Journ., vol. x.). It may be shown by means of a transformation of the series for Con- , sin x , , , x cos x and that tan x= x 2 a? y? m.- T- i tinned This may be also fractions 1- 3- 5- 7- " easily shown as follows. Let j/=cos /x, and let y , y" . . . denote . the differential coefficients of y with regard to x, then by forming caTfunc- these we can show that 4xy"+2y + y=Q, and thence by Leibnitz s . theorem we have Therefore .= -2 -f^= " " ^ y y/y y( n + l ) hence -2VxcotVx= -2- -6- -10- -14- " Replacing Vx by x we have tan x= = ^ - Euler gave the continued fraction _n tan x (ri* - 1 ) tan 2 x (7i 2 - 4) tan 2 x (n? - 9) tan 2 x Lclll ?ZC , J this was published in Mem. de I Acad. de St Petersb., vol. vi. Glaisher has remarked (Mess, of Math., vol. iv. ) that this may be derived by forming the differential equation where ?/=cos(rearc cosx), then replacing a; by cos x, and proceeding as in the former case. If we put 7i = 0, this becomes ^_tan x tan 2 x 4 tan 2 x 9 tan 2 x whence we have arctanx- 1+ ^ ^ 7+ ... It is possible to make the investigation of the properties of the
simple circular functions rest on a purely analytical basis. The sine
