lo Mr. Herschel on a remarkable Application 



Before we proceed to the application of these transformations, 

 it will be necessary to premise some properties of the func- 



tions K and x'. 



— I / —I N _.-.l / —I 



Let a= cos. • (^ ) and a! =• cos. e. 



* 



• This notation cos. ~^ e must not be understood to signify , but what is 



■° ^ COS. e 



usually written thus, arc (cos. rze). It is true that many authors use cos." A, Sin." 

 A, &c. for (cos. A)", Sin. A"" ; lest therefore the notation here adopted should appear 

 capricious, it will not be irrelevant to explain its grounds. If q) be the characteristic 

 mark of an operation performed on any symbol, x, (p(a:) may represent the result of 

 that operation. Now to denote the repetition of the same operation, instead of 

 <P{<P{x)) ; <p(<p(?'(^))) J &c. we may most elegantly write <p^{x); <p*(x); &c. Thus we 

 use d^x, A^.r, IL'^x, for ddx, AAAj:, SIjt, &c. By the same analogy, since sin. x, 

 cos. X, tan. x, log. x. Sec. are merely characteristic marks to signify certain alge- 

 braic operations performed on the symbol x, (such as 



\ I ^ 1.2 1.2.3 ' 3 (. ^ I ^ 1.2 ^ 1.2.3 J 1 



&c.) we ought to write sin. *x for sin. sin. x, log. ^x for log. log. log. x, and so on. 

 To apply this to the inverse functions, we have (p" (p" {^x) = (p"+"' {x) . Hence if 

 m zz — n (p" (p—" {x) =z cp" {x) =. X. with the operation ((p) performed no times on 

 it, or merely x, that is, <p— " (x) must be such a quantity that its nth {p) shall be x, 

 or in other words <p"~"(^) must represent the nth. inverse function. It frequently 

 happens that a peculiar characteristic symbol is appropriated to the inverse function. 

 Let it be iV> then 9~", x z=. M* ^> and (^"x zz (p*", (p"-", x = (p" (<p", 4'";. x), hence 

 •^"> <f>"> X zzx, and therefore ■^—" x rz ■^~~" ■>]/" p", x zz ^+'', x. For instance d""" 

 V =i/"V, d'Vzz V . — 2—", X :=. A"x . a" =: i, with the operation of multiplying by 

 0, n times performed on it, and .'. .a"~" zz i, with the inverse operation so often per- 

 formed on it, =: — ; a'zzi. Similarly sin.""* x ~ arc (sin. = x) . cos. — * j: =: arc 

 a" 



(cos. * == jr) &c. — and ifczriH j 4-&c. r=: log. c* and .•. c' = log. * x. 



c (n)' 



c zz log.—* X, and c = log.-~''r, or the nth inverse logarithm of :«■. It is 



easy to carry on this idea, and its application to many very difficult operations in the 

 higher branches will evince that it is somewhat more than a mere arbitrary con- 

 traction. 



