Suppose that there are equal roots in the above equation, say three roots equal to a. The resolution of the fraction gives terms of the form E(n - + L(D - which contribute to the general value of y, e"" + + Mfd.e and the arbitrary part ea (e+ ex+ The linear equation of differences corresponding to the above is + + x, where w, is a function of .r to be determined. The operation per formed on ts, on the first side is + Every single root a, contributes to the solution a term of the form A (u - x or x), in which x may be any function of which the difference is If .r be an integer, what is called 1 will do. Any set of equal roots contributes terms of the form A(E - x or Any linear equation being Oven, in which either of the operations E„ A or D, is combined with either E„ D,,, or 14, the form of the solution may be found. Take for example of 7; - aos.s+i =the operation performed upon it,,, on the first side is to, - an" and accordingly we have xdx, which is one form of the solution, and must be interpreted by expanding into 1 axE, + 2 Another form can be obtained from 1 (E, - ) = - X.
We can only touch very briefly upon these points, and rather to low the existence of the system than to enter into it. Further details will be found in the Library of Useful Knowledge, in the Treatise on le Differential Calculus,' pp. 751-758.
The theorems answering to that of integration by parts, when D -a Luc! are used, are as follows. To save room let v-a and E-a se denoted by 8 and A. Then (m) = = I and Q being functions of .r, to which to and E refer, and V meaning
/p: d.r. If a = 0, the first becomes roailx = gdx — f f telt , which is the formula for integration by parts. And if Q be of the 'cm 8•11, or Ion, and r be a rational and integral function of a lower legree than the nth, the preceding operations carried on will show that Rand (PA • R) can be performed without leaving any trace rf inverse operation in the result. Of the first of these it is a par icular case that d" dx an be found whencyer r is a rational and integral function of a lower legree than the nth. Thus, P being of the second degree, r"On By help of these theorems the intermediate equations of any linear nuation can be readily discovered. Suppose, for instance, we have (rt - x, a equation of the eighth degree. There are eight equations I the seventh degree. Two of them are discovered at onco by oerforming the operations (v -1 and (n - 2 on both sides, lying (n - (n - y = xd.r, • (o (D - xel.r.
To find the other six, multiply separately by x, the implest functions of their several degrees, and perform ( n tpon all four results, and (n- upon the first two. This, by he preceding theorems, can be done. Thus, multiplying by x we Ave (D-1? I 2py- (so -2)y = - y = .rxsix, wish are two more of the required equations. To find the equations I the sixth degree, those of the seventh degree must be selected hich admit of a repetition of the operation without leaving the inverse form (to or (n : and the operation must bo repeated ; and so on.