of certain Partial Differential Equations. 215 



the former or the latter to be used according as t"^ is greater 

 or less than or. 



Of course, to obtain ?« + C from these, we have still to inte- 

 grate once with regard to t. Nevertheless j- is an integral 



of the proposed equation ; but it is not the complete integral. 

 We have also to notice that, when this integration has been 

 effected, we must then write in the resulting integrals t + h 

 for t. 



Thus integrals of equation (1) have been found in finite 

 terms when a is any integer whatever ; but in the remaining 

 case, when a is not an integer, we have only found the general 



value of -J-, which, however, is itself an integral in finite terms. 



I will now turn to another general class of differential equa- 

 tions which has hitherto resisted all efforts to find a general 



integral of it. If we write .v^-a for x, and ^ for (1— <2)Hn 

 equation (1), the result will take the following general form, 



w^'^^-'d?' (^> 



which will be at once recognized as an equation which has 

 never been integrated in finite terms, except in a certain class 

 of particular cases. Its integral may be deduced from the 

 results contained in II. and IV. by making the forementioned 

 substitutions for x and t in them. 



There is yet one other differential equation which is men- 

 tioned by the Astronomer Royal as being of importance in a 

 certain problem of sound, and has not been integrated in finite 

 terms. It is of the follow^ing form, 



d\i d}u , , du .„. 



W=M-^^^d-x (9) 



By the method of germ-integration the root-integral of this 

 equation is found to be 



+ C={-^i + -^i}e-«(^^^±^-~)^; (10) 



^{os-\-ty {x—ty J ^ ^ 



and this root-integral will become the general integral of the 

 equation (9) if we write t + h for t, and x + k for x in it, h and 

 k being the germs of t and x respectively. 



ShefEeld, August 17, 1877. 



u 



