214 The Rev. S. Earnshaw on the Finite Integrals 



is an unsatisfactory form of an integral when its convergence 

 is not assured. 



Now on looking at the above equation, we see that mt + li, 

 •±imx, and z^ + C may be respectively written for t, x, and u 

 without in any way affecting it — A, w, C being what I have 

 elsewhere called ger7ns, i. e. quantities capable of being con- 

 sidered either arbitrary constants or arbitrary independent 

 variables. Hence we know that the integration of equation 

 (1) comes within the Germ-integral Theory ; and by this theory 

 the following results have been obtained: — 



I. If a= 2n + &, ^ being an integer, and if lu be the integral 

 of equation (1) when h is written for a, we have the following 

 important formula, 



"+'^=(:;^.)"'" (^) 



II. Hence, when a is an even integer and 5 = 0, we have the 

 following integral of (1) for this case, 



a 



(It will be perceived that Y{x±t) is used as a brief represen- 

 tative of the sum of two independent arbitrary functions, 

 Y{x-^t) and/(.'^-^).) 



III. If a be an odd integer and 5 = 1, the integral of (1) 

 will in this case be 



when t'^ is greater than x^; but when t'^ is less than x^, then 

 the integral takes the following form, 



z, + C= ('-4-y"^|Acos-'-+Bsin-i-j. . . (5) 

 \xdx) i. X xj ^ ^ 



But as these are what in the germ-theory are denominated 

 root-integrals, it will be necessary, in order to convert them 

 into general integrals, to write in them t + h for t, h being ^'s 

 germ. 



IV. When a is not an integer, we find 



~ =A(t''-x')-l-hB{t''-x')l-'x'-, . . (6) 

 or 



^-' = Av^^2-^2)-^ + B(.r2-^2)^-V-«, . . (7) 



