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as the constant is arbitrary.

As operating factors of the form + n° very frequently occur in differential equations, it is convenient to keep in mind that the complementary function due to it is of the form C cos nx + C sin na.

(8) Let * + n°y = cos mæ.

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The roots of the equation

geen – a" = 0, are included in the formula

a {cos P +(-)! sin p}, where p=4*, , receiving all values from 0 to n; and the roots corresponding to 1 = 0 and 1 = n being + a and a.

If now we take a pair of the impossible roots, which we may call a and ß, the corresponding terms in the general value of y are

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But by the theory of the decomposition of rational fractions we know that N

- cos 0 + (-): sin 2na?n

1 2naan 2na?" - 1

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G -B) X = ear cos $ {cos (a x sin o) - (-)} sin (av sin o)}

x de Xe-ar cos + {cos (ao sản p) + (-) sin (a sin ¢).

Therefore substituting these expressions, the two terms in the value of y become after reduction,

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Also the two roots + a and – a give rise to the terms

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Ilence we put the expression for y under the form

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Sdx { X € -1,* cos sin (a x sin o)}]. The symbol I implies the sum of terms derived from assigning to p in the preceding expression all values from

to (n − 1).1.

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The complementary functions are, for the sake of shortness, supposed to be included in the signs of integration ; but if we wish to see their form, we have only to make X = 0 in the preceding expression when it becomes y = Ce-ar + C."*

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COSI a 2

(Tsin

. 27 277

27 2T sin — +- +

- tn nl

nn + &c.

+ &c.
This is evidently the solution of the equation

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Euler, Calc. Integ. Vol. 11. Sect. 2, Cap. iv.

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n being a positive integer.

Here the operating function is (a + b)", which is composed of n equal factors ; consequently y= (a + ) * = 6-4+ /* dx" "X+8~* (Co+C, « + Cox® + &c. + Cn-,*).

The term €-4* /* dx" c^* X may either be integrated by successive steps, or by the general formula for integration by parts; or what will generally be more convenient, the function (a + ) may be expanded in ascending powers of .

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If n were negative or fractional, the first term would retain the same form, but the form of the complementary function would be different from the difference between the roots of

(x + a)" = 0, when n is integer and when it is fractional or negative. I cannot however here enter into a discussion of the difficulties of this subject, which is closely connected with that of General Differentiation. Euler, Calc. Integ. Ib.

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(18) Let

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dan +

dil Qe. +

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