f(x) = (-)-1 1.2.3... (n − 1) (1 + x)-", x)−", fr (0x) = (-)-11.2.3... (n-1) (1+0x)" ; ƒ" (0) = – 1, = 1.2, ƒ'''(x) = (1−x2)− § + 3 x2 (1 − x2)—§, f(x) = 9x (1−x2)− +3.5 x3 (1 − x2)−1, * ភ្នំ The general derived-function is of too complicated a form to ƒ""(0) = 1, ƒ'(0) = 0, f'(0) = 9. But sin-1 may be more easily expanded by the following artifice, which adapts itself to those functions of which a derived-function assumes an algebraical form. Differentiating equation (8), Art. 57, we have X3 ƒ'(x) = ƒ'′(0) +ƒ”(0) +ƒ'''(0) +f'(0) + x2 1.2 1.2.3 whence, equating coefficients of the same power of x in the two series, we have sin-1 x = x + + 1.3.5.7... (2r-1) 1.2.3... (2r-1) 2r 2r 1.2.3... (r-1)r 1 1.2 x3 1.3 1.2.3.4 x5 1.3.5... (2r-1) 1.2.3.4... (2r−1) 2r 2r ; -1 cos-1x, tan-1x, log (1+x), are other functions which may be conveniently expanded by this method; and which the student is recommended to apply the process to. 60.] As many properties of some series which have been expanded in the last article will be required in the sequel of the Treatise, it is most convenient to introduce them here, though they may more properly be considered to belong to analytical trigonometry. By an imaginary or impossible quantity is meant, one of the form, a and b being symbols ties, and the symbol a+b√ −1 ; (23) of positive or negative possible quanti-1 being that, which, when squared, is equal to -1. Two such expressions, which differ only in the sign of √1, are said to be conjugate to each other; thus a+b√1 and a-b√√-1 are called conjugate imaginary expressions; and it is to be observed, that the product of two such conjugate expressions, viz. (a+b√1) (a−b √−1) = a2 + b2. (24) Now such an expression as (23) may always be put under the form r (cos + √1 sin 0); (25) in which caser is called the modulus of the expression a+b√1. For let and as a and b are possible quantities, a2+b2 is a positive quantity, and therefore r is possible; and as tan 6 passes through all values, from through 0 to +∞, as increases from π П 2 to + whatever are the relative signs and magnitudes of a 2' π and b, there is always some angle between and b 2 2 π which will satisfy the equation, tan =; therefore the substitutions 0 made for a and b are possible, and therefore a+b√-1 may always be put in the form r (cos + √1 sin 0). Also, 61.] In series (15), Art. 59, successively write for x, x √−1 and -1; then and taking the Napierian logarithms of both sides of the equation, we have 2x√1 = log (1 + √−1 tan x) — log (1 - √1 tan x) the series being expanded by equation (18), Art. 59; whence, equating impossible parts, and dividing both sides by 2√—1, we have which is a series useful for the calculation of π. Again, by equation (31), = cos x + √1 sin æ ; (34) .. e√ = cos y + √ −1 sin y; therefore by multiplication, e(x+y)√=ī = cos x cos y — sin x sin y but e(x+y) √=1 = cos(x + y) + √1 sin (x+y); +1 (cos x sin y + cos y sin x), (35) (36) wherefore, equating the possible and impossible parts of the equal quantities (35) and (36), we have the fundamental trigonometrical formulæ, cos (x+y)= cos x cos y sin x sin -- (37) (38) Again, (cos x + √−1 sin x) (cos y + √—I sin y) (cos z + √−1 sin z) ..... = cos(x+y+z+ ...) + √√−1 sin (x+y+z+...)...; (39) whence, if x = y = z=... to m quantities, (cos x + √1 sin x) = cos mx + √−1 sin mx ; x mx which is De Moivre's Theorem. (40) By these processes therefore the multiplication of a series of factors of the form cosx+1 sin x, and therefore of all imaginary expressions, is reduced to the addition of the arcs under the circular functions; and the involution of such quantities to the multiplication of the arcs. 62.] Equivalent expression of (cos x)", in terms of the cosines of the multiple arcs. To abbreviate the notation, let us substitute as follows: |