Starting from these values and giving p successively all integer values from 0 upwards, we find

When n is an even integer the second line, being mul

π

tiplied by the cosine of an odd multiple of vanishes, and the

2

first line alone remains: when is an odd integer the first line vanishes and the second line alone remains. When n is a fraction both lines must be retained, except for some particular values of n which cause the factor of one or other series to vanish.

find

(7) To expand sin næ in ascending powers of sin . Proceeding in the same manner as in the last example, we

When is an integer the first series always vanishes, and the second is positive or negative according as (n - 1) r is even or odd. When n is odd the second series terminates; when n is even it continues to infinity. When n is fractional both series coexist, except for particular values of r.

(8) To expand cos ne in ascending powers of sin a, and sin næ in ascending powers of cos æ.

Proceeding as in the last two examples, we find

When n is an integer the second line always disappears, and the first series terminates when n is even, and does not terminate when n is odd. When n is fractional both series are retained, except for particular values of r.

When n is an odd integer the first line, when ʼn is even the second, alone remains; but when n is fractional both series are retained except for particular values of r. In no case do the series ever terminate.

For an exposition of the difficulties concerning these expansions, and the discussions to which they have given rise, the reader is referred to Poinsot's Memoir on Angular Sections, where the complete form of these expansions was first given.

If we were to endeavour to effect this by means of Maclaurin's Theorem, we should find that all the differential

odd powers of x above the first; for if we assume

and comparing the coefficients of like powers of x,

B1, B3, &c. being coefficients to be determined.

If therefore C2, be the coefficient of " in

the equation

n

2

and if in Ex. (27) of Chap. 11. Sect. 1, we make r = 2ǹ 1,

12n-1}

1

1-1} - &c.].

2n

1

Ban

+ {32n-1

2n
1

22n-1

2n (2n-1)

12n-1 - &c.].

1.2

These coefficients B,, B... B-1, are of great use in the expansion of series, and bear the name of Bernoulli's numbers, having been first noticed by James Bernoulli in his posthumous work the Ars Conjectandi, p. 97; but the complete investigation of the law of their formation is due to Euler, Calc. Diff. Part II, Cap. v.

(10) To expand tan by means of the numbers of Bernoulli

The coefficient of 0-1 in the expansion of this function

will be the same as that of " in the development of € + 1 multiplied by 22" (-)". By what has preceded it appears, therefore, to be equal to

(11) To expand cot by means of Bernoulli's numbers.

Now the coefficient of 02"-1 in this expression is the same