In this case, we may make r equal to zero in both members of the equation: or, in other words, we may replace c, by co and s, by So, If 2 cos 0 be positive and m even or odd p = c. If 2 cos 0 be negative and m even If 2 cos be negative and m odd In all these cases, the value of p is expressed, therefore, by a series of cosines, the corresponding series of sines being equal The terms of the series for c, being, therefore, the same from the beginning to the end, it follows that m (m-1) c。=2{cos me+m cos (m − 2) 0 + cos (m −4) 0 + •} 1.2 2 m+1 terms, m continued to +1 terms, when m is even, and to when m is odd: the last term, in the first case, being which is a modified form of the half of the middle term of the binomial (1+1)" (Art. 489): and in the second case being In both cases, therefore, p is expressible either by a series of cosines or of sines, unless cos 2mr=0 or sin 2mr=0 in one case, and cos m (2r+1) π = 0 or sin m (2r+1)π = 0 in the other. In examining such cases, we may suppose m a rational fraction in its lowest terms of the form P, and also that r does not exceed n-1. n If p is odd, n divisible by 4 or pariter par (Art. 516, Note), In these cases p is expressible by a series of sines only, and be of the form 4i+1 or pariter impar, and p=s, or if Р if p be of the form 41+3 or impariter impar. n Secondly, if r=0, or if p be odd, n even and r= = we find 2 In these cases, p is expressible by a series of cosines only, and p = c。 or -C Thirdly, if p be odd, n even and of the form 41+2 or im In these cases, p is expressible by a series of sines only, and p = − S3n-2 or S3n-2, according as p is pariter or impariter impar. In these cases, p is expressible by a series of cosines only, or C-1 according as p is even or odd. and 1058. Again, let it be required to find a series for (2 sin 0)", in terms of the sines or cosines of multiples of 0. Since 2-1 sin 0=e°√=1_e-0√1 (Art. 926), we get (2√=1 sin 0)TM=1′′ {em®√=1 _ mem-2) 0√√=1+ m (m−1) 。(m−4)0√=1_...} if cr 1.2 = cos m (2r+0) —m cos (m − 2) (2rπ+0)+ √=1{sin m (2 rπ + 0) — m sin (m − 2) (2 r π + 0) + = c, +8,√-1, ... •} be taken to represent the series of cosines, and s, the series of sines. If ρ if 2 sin denote the arithmetical value of (2-1 sin 0)", then, {cos m (2r+4) π + √−1 sin m (2r+ 4) w} p = c,+s,√1: and if 2 sin be negative {cos m (2r+†) π + √−1 sin m (2r+})} p = c,+s, √−1. 1059. If m be a whole number and even, then 2-1 (sin 0){cos me — m cos (m − 2) 0 + . . . (2 sin 0)". When m is a whole number. m+1 to ... terms}, 2 the sign or being used, according as m is of the form 4i+1 or 4i +3. 1060. If m be not a whole number and 2 sine positive, then When mis or its value is expressible either by a series of cosines or sines of multiple angles, except under the following circumstances. If m2 (in its lowest terms), p and n being odd numbers, then = n according as p is of the form 4i+1 or 4i + 3. If p be even and n of the form 4i+1, then according as p is of the form 4i or 4i+2. If m be not a whole number and 2 sin 0 negative, then its value being expressible, either by a series of sines or of cosines, except under the following circumstances. If m=P (in its lowest terms), p and n being odd numbers, n then when n is of the form 4n+3, the sign + or being used, according as p is of the form 4i +1 or 4i +3 in the first case, or the contrary in the second. If p be even and n of the form 4i+3, then the + or sign being used, according as p is of the form 4i or 41+2 in the first case, or the contrary in the second. We have been thus minute and critical in the deduction of all the separate cases which this problem comprehends, not only on account of the intrinsic importance of the problem itself, but likewise as affording a very instructive example of the proper mode of discussing and interpreting a formula when it is expressed in very general terms: for it will very generally be found, that the more comprehensive is the form in which a problem is stated and investigated, the more remote and difficult its application will be to the particular cases which it includes. been Series for cosmo cos me and sin me, in inves- terms of cos or A converse problem to the one which we have just considering, would require us to assign a series for and sin me in terms of the sines and cosines of 0: the tigation however of such series, which branch out into a variety of forms and cases, cannot easily be effected without the aid of principles and processes which will be given in a subsequent volume of this work. great sine. as distin from Arith 1061. The same principles will likewise find their appli- Theory of cation in the general theory of symbolical as distinguished from symbolical arithmetical logarithms: if we assume p to represent the arith- guished metical logarithm of a or of its powers, we may extend our metical enquiries to determine the most general symbolical forms of the logarithms. logarithms of (1 x a)", of (-1x a)", or of 1 x (1 x a)", or, in other words, we may suppose a or a" to be affected by any sign, which is recognized in Symbolical Algebra, and their logarithms. to be required, assuming e to be the common base to which they are referred. and (−1) = cos m (2r+1) π + √−1 sin m (2r+1) π = em (2r + 1) = √=1 ̧ it will follow, in conformity with the definition of Napierian logarithms (Art. 901) that The logarithm of 1" is zero, when m=0 or r=0: the logarithm of (-1)" can only become zero when m=0 and therefore (-1)=1: the other logarithms are all imaginary, and are unlimited in number: it is the logarithm of 1° or (-1)0 only, which is essentially zero and which admits of no other value. 1062. If m be a fraction in its lowest terms, with an even denominator of the form, there is one value of 1" which is 2n equal to -1, corresponding to r=n: in this case we find The Napierian Jogarithms of 1" and (-1)". Case in which one |