91 

 equation (1) becomes a complete determinant when multiplied 



by e-^^"''- 



The problem of expressing the coeflScients of the differen- 

 tial equation (2), in terms of its particular integrals, has 

 been treated by M. G. Libri, in a very elegant memoir on 

 Linear Differential Equations, printed in the tenth volume of 

 Crelle's Journal. He has given the following formula to 

 determine A, : 



(n-2)Z)2 

 D 



l^©- 



and merely indicated the method of obtaining expressions for 

 the other coefficients. From the nature of this method, how- 

 ever, it is easy to see that it would be scarcely possible to 

 write down the values of the higher coefficients, in terms of 

 tto, Ml, &c., on account of their extreme complexity. M. 

 Libri has noticed that the expression given above for A^ is, 

 from the nature of the case, a symmetrical function of Wo> "u 

 &c. ; though this is not indicated by its actual form. To ex- 

 hibit it as a symmetrical function of those particular integrals 

 we must execute in it all their possible permutations, and 

 then take the sum of the results. This operation consider- 

 ably increases the complexity of the formula. 



[In the notice of Mr. Donovan's Lamp (p. 75), it was 

 omitted to be stated that it burned with a brilliant light dur- 

 ing the sitting of the Academy.] 



i2 



