And as ei«+(-): B!* = fax {cos Br + (-) sin ßx}, and -(-)!$;" = far {cos Br - (-) sin ßx}. The sum of the two corresponding terms in (6) is 26"* (A cos Bir + B sin Br) sdx (e - ax cos ßx. X)) A’ + B? Į (9). + 2€«X (A sin 3x - B cos ßx) dx (e-a x sin ßx. X) A + B2 B (10). (1? + B2) The sum of the complementary functions {C, c(-) BX + C, €-(-)! Be}, may evidently be put under the form Fax (C cos Bic + C' sin B.1') = C'eak cos (B x + a). (11) If there be a number of equal pairs of impossible roots in the equation (3), the general expression for the value of y becomes so complicated as to be of little use, and it is therefore unnecessary to insert it here. The preceding process may frequently be simplified in its application to particular cases, by means of the following considerations. The inverse operations are always reduced to the sum of several of the form Y where r may be 1 or any positive integer. Now this operation will have a different effect according as it is expanded in ascending or descending powers of , that is, according as it is considered to be inasmuch as in the one case it will involve integrals, while Term as in the other it will involve differentials only. But a simple relation connects the two, for (a + b)*x + ( a + a) *0. Now the latter term ( + a) "* =--** da" o =e-ar (Co+Cjx +Cgx® + &c. + C...er=), and the former term (a + m) X, being expanded in ascending powers of our will give rise to a series of differentials which are always easily found, and which, when X is a rational and integral function of æ of n dimensions, always breaks off at the (n + 1) th term. But since each factor of the form gives rise to a separate complementary function, while X is operated on by all in succession, it is sufficient to expand Jy in descending powers of X, without splitting it into its binomial factors, and then to add the complementary functions corresponding to each of these factors. If the function X be of the form ems, the result of the operation f 6 ) <** takes a very simple shape. For if we expand $ (.) in ascending powers of hy so as to have a series of the form {A+B +C ( + )*+&c.}eos, and then operate on em* with each term separately, we find, as a ) <** = m" ", that the series becomes {A + Bon + Cm? + Dm+ &c.} {** = f (m) ***. a very Hence for example, we have T (m - a)". The preceding theory may be stated in the form of the following proposition : if the integral of an equation 1 1 1 3 11 d.) } + Cellt, XL a \d x) a? \d x) '03 d. the differentials after the fourth being neglected. Effecting the differentiations x* 4 X 4.3.2.2 4.3.2.& 4.3.2.1 therefore by the formula (11) y = C cos n x + C'sin n x = C, cos (n & + u). The same result may be obtained by a different process, which is subjoined as it points out very distinctly the reason why these circular functions appear in the integral. Id2 -1 1-2 + n'! (= It is indifferent in which order we perform the operations ; |