A Commentary On Newtons Principia With a Supplementary volume volume 2

J M F John Martin Frederick Wright

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3 ____ i / R 1 In like manner, the coefficient of -, . , m . M cos. N (* & } in the r + i function (f) is 2. 1. 3. 5 ____ (i + n) . 1. 3. 5 ---- (i n) 2. 4. 6. (i + n 1) . 2. 4. 6 (i n 1) this is the coefficient of m m in the value of /3, when we neglect the squares of m, m, and when i n is odd. Comparing this with the va lue already found, we shall have > /I- 3. 5 (2i IK* i(i l).... (i n + 1.
V V 1. 2. 3 i ) (i+1) (i + 2) (i + n) an expression which is the same as in the case of i n being even.... If n = 0, we also have /I. 3. 5.... (2 i- IV 7 \ 1. 2. 3 i ) BOOK I. ] NEWTON S PRINCIPIA. 241 560. From what precedes, we may obtain the general form of functions Y w of m, V 1 m 2 . Sin. -a, and V 1 m 2 . Cos. R, which satisfy the equation of partial differences = - - .
\ dm / 1 m 2 Designating by /3, the coefficient of sin. N &, or of cos. N ^, in th function Y (1), we shall have - - r - -T. .
dm 1 m 2 8 is equal to (I m 2 ) & multiplied by a rational and entire function of m, and in this case, by the preceding No.

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