REPORT ON CERTAIN BRANCHES OF ANALYSIS. 221 



which is adapted to the immediate apphcation of the general 

 principle in question. 



Thus, ii' u = e'" *■, we get 



d x-"^^ ' d x-" "^ "" d X" ~ "* ^ ' 



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

 quantity whatsoever. 



it u = sm m X, -J— = m sm {-—■+mx), -z — -„ = 7n^ sin 

 dx \2 / d X- 



d^ U ( T "K \ 



(w + m x), . . . . y—y = m'' sin ( —^ + m x j when r is a whole 



number, and therefore generally. In a similar manner if 



u = cos m X, or rather ti — cos m {Vf x, (introducing P as 

 a factor in order to express the double sign of m x, if de- 

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



fjh' -ft \ T TT I 



- — = (m a/ ly COS < —^ + {m \^ I) x >, whatever be the 

 value of r. If u = e" * cos m x, we get, by very obvious re- 

 ductions, making o = . - and fl = cos ' — , 



-, — = p'' e"'^ cos (m X + n fl). 

 dx^ '^ ^ ' 



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



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

 member of this equation, we shall get four values of — ^°^ *" ^, instead of 

 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 cos m x ^ j^ ^j^.^ 



dr 

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



aJx 

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



and its equivalent series 



x^ + 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 continue 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 difteretit in those cases where the several terms exist indepen- 

 dently of any explicit or implied process of derivation. 



