SECT. IV.] DIFFERENTIATION OF FUNCTIONS. 437 



422. We may also remark that we can deduce from equation 

 (B) a very simple expression of the differential coefficient of the 



d l [* 



i th order, -T-j/OOi o r of the integral I dx l f(x). 



The expression required is a certain function of x and of the 

 index i. It is required to ascertain this function under a form 

 such that the number i may not enter it as an index, but as a 

 quantity, in order to include, in the same formula, every case in 

 which we assign to i any positive or negative value. To obtain it 

 we shall remark that the expression 



cos 



^7^ . ITT 



or cos r cos -^ sin r sin -=- , 



4 A 



becomes successively 



- sin r, - cos r, + sin r, + cos r, sin r, &c., 



if the respective values of i are 1, 2, 3, 4, 5, &c. The same results 

 recur in the same order, when we increase the value of i. In the 

 second member of the equation 



cos x ~ 



we must now write the factor p* before the symbol cosine, and 

 add under this symbol the term -f- i- . We shall thus have 



The number i, which enters into the second member, may be 

 any positive or negative integer. We shall not press these applica 

 tions to general analysis ; it is sufficient to have shewn the use of 

 our theorems by different examples. The equations of the fourth 

 order, (d\ Art, 405, and (e), Art. 411, belong as we have said to 

 dynamical problems. The integrals of these equations were not 

 yet known when we gave them in a Memoir on the Vibrations of 



