1871.] Mr. W. H. L. Russell on Linear Differential Equations. 281 



Feburary 2, 1871. 



General Sir EDWARD SABINE, K.C.B., President, in the Chair. 

 The following communications were read : — 



I. « On Linear Differential Equations.— No. IV." By W. H. L. 

 Russell, E.R.S. Received November 17, 1870. 



I will now consider some interesting results in definite integrals obtained 

 from the solution of linear differential equations. 

 Let us first consider the linear differential equation 



te+ w + («. + % + («, + k«=) % + («c + Wy=». 



where 6 3 =1. 

 Let 



where the limits are to be determined. Substituting in the differential 

 equation, and following the usual method, we find 



Now let 



a 3 (iz) 3 + a 2 (iz) 2 + « t (fe) + a ,_ m A _ a* _ y 

 ft 8 (iar) 8 + b a (iz)* + & A (uO -f £ w-a iz—p iz—y 



then we shall have 



_ C e^+^.diz 

 V ~ J(iz—a) x+1 (iz—py + \iz--yy +l$ 



where the limits, determined by the usual method, are evidently — oo and 



+ 00. 



. We have, from the above equation, 

 a 3 =b 3 m } 



a 2 =-h 3 {\ + fi + v + m(a+fi + y)}, 



a=b/ i X(i3 + y)+fi(cc + y) + y(a + fi) + m(a(3^ay+M}^ 

 a o= — b 3 {^py + pay + m/3 + ma(5y} } 

 ^=-^( a + /5 + 7 ), 



b =—b 3 a(3y. 



VOL. XIX. Z 



