802 



A paper was also read, entitled "On the Integration of Linear 

 Differential Equations." the Rev. Brice Bronwin. Communi- 

 cated by C. J. Hargreave, Esq., F.R.S. 



The method chiefly employed in this paper, is analogous to one 

 which the author had previously applied (Camb. Math. Journal, 

 No. 4) to the integration of such equations in cases where the co- 

 efficients are integer functions of the independent variable. Here 

 they are any functions of that variable, it being however understood 

 that in all integrable cases there must be some relation among these 

 coefficients. The integration is effected by a general theorem of 

 the form 



where D denotes any function of x, and ■or a function of symbols 

 both of operation and quantity. By means of this theorem, and the 

 substitution u = 'Zu^ ixr, . . . im,2 v, or some other similar one, the equa- 

 tion is either reduced to an integrable form, or to an equation of a 

 lower order ; or, when neither of these objects can be accomplished, 

 the method may be employed to effect a transformation. 



The method applies most readily to equations of the second order; 

 but may be applied to those of a higher order, the coefficients be- 

 coming more restricted as the order rises. The integrable cases are 

 very numerous and vary considerably in form ; and, as each distinct 

 form requires a variation in the process, they are distributed into 

 classes. In each class, a few particular examples, derived from the 

 general cases, are given. 



By means of the general theorem, the equation 



zu,r:'UT,i u-j-ppu = X 



may be integrated in the most general ca^e, or when the coefficients 

 are any functions of i\ having, however, certain relations between 

 them. 



Several theorems of the form rnpit — piTn-i^f, where p = 'D-^Q, 

 ::'« = D2 + A„D-fB„, or similar to it, are given. They are not found 

 without difficulty ; are much more restricted in their application 

 than the general theorem ; and lead to but few results ; but they 

 are deserving of notice on the ground that they may possiblv suc- 

 ceed in a particular case when all other methods fail. 



A few general examples of a class of equations, the solution of 

 which is attended with considerable difficulty, are next given. These 

 are of the forms, 



and others varying a little from them. 



The concluding part of the paper is occupied with the transfor- 

 mation and application of one or two of the general theorems which 

 have been given by the author in the Cambridge Mathematical 

 Journal, Xe^^ Series, vol. iii., from which a few examples, more or 

 less particular, have been derived. 



