REPORT ON CERTAIN BRANCHES OF ANALYSIS. 221 



which is adapted to the immediate apphcation of the general 

 principle in question. 



Thus, if M = e'""", we get 



d X d x'- " ^ 



when r is a whole number, and therefore, also, when r is any 

 quantity whatsoever. 



If « = sin mo:, II = «^ sin [j + nixj, ^, = m^ sin 



(^ + «^a:), .... ^^; =^'-sin(^ + ^»^)whenrisawhole 

 number, and therefore generally. In a similar manner if 

 « = cos m X, or rather u ^ cos m {Ifx, (introducing P as 

 a factor in order to express the double sign of »* ^, if de- 

 termined from the value of its cosine,) then we shall find 



^ = (m a/ l)*- cos 1^ + {m x' 1) ^'j^, whatever be the 

 value of r. If u - e« ^' cos m x, we get, by very obvious re- 

 ductions, making p = . ^ and 9 = cos — , 



d' u 

 dx^ 

 It is not necessary to mention the process to be followed in ob- 



dx^ "^ 



and if we combine arbitrarily the double values of the two parts of the second 



1 i^ d' cos mx . , J » 

 member of this equation, we shall get four values of ; — , instead ot 



two ; and, in a similar manner, if we should resolve cos m x into any number 



of parts, we should get double the number of values of ^ If this 



principle of arbitrary combinations of algebraical values derived from a. com- 

 mon operation was admitted, we must consider j— — ^ as having two values, 



and its equivalent series 



x^ + x'^ + ** + &c. 

 as having an infinite number. But it is quite obvious that those expressions 

 which involve implicitly or explicitly a multiple sign must contmue to be 

 estimated with respect to the same value of this sign, however often the reci- 

 pient of the multiple sign may be repeated in any derived series or expression. 

 The case is different in those cases where the several terms exist mdepen- 

 dently of any explicit or implied process of derivation. 



