SECT. IV.] SYMBOLICAL METHODS. 403 



This known abridged notation is derived from the analogy 

 which exists between integrals and powers. As to the use made 

 of it here, the object is to express series, and to verify them 

 without any development. It is sufficient to differentiate under 

 the signs which the notation employs. For example, from the 

 equation v = e tl} * &amp;lt;f) (a?), we deduce, by differentiation with respect 

 to t only, 



which shews directly that the series satisfies the differential 

 equation (a). Similarly, if we consider the first part of the series 

 (X), writing 



we have, differentiating twice with respect to x only, 



Hence this value of v satisfies the differential equation (a). 



We should find in the same manner that the differential 

 equation 



gives as the expression for v in a series developed according to 

 increasing powers of y, 



v cos (yD) $ (x). 



We must develope with respect to y, and write ^- instead of 

 D : from this value of v we deduce in fact, 



? = --D COS 



The value sin (yD} ty (x) satisfies also the differential equation; 

 hence the general value of v is 



v = cos (yD) &amp;lt; (x) + W, where W= sin (yD) ty (x). 



262 



