REPORT ON CERTAIN BRANCHES OF ANALYSIS. 261 



sequeiitly, in the first case we have <p = /i = <Pi : 

 in the second, 



9 =/, +/2 = ?2, and therefore/^ = (^2 - ^, : 

 in the third, 



<P =fi +A+f3 = <P3> and therefore /a = (p^ - ^3. 

 It appears therefore that 



is a formula which is competent to express all the required 

 conditions of discontinuity *, 



Equivalent forms may be considered as permanent within 

 the limits of continuity, and no further, unless the requisite 

 signs of discontinuity, whether implicit or explicit, exist upon 

 both sides of the sign = : thus, the equation 



4 r ^ 4 6 "I 



•'■D/cos^ = —\ T-^ sin 2 a; + -^^— psin4^ + -=— =sm6a: + &c. \ 



11.0 0.0 0,1 I 



is permanent within the limits indicated by the sign '*'D^° and 

 no further, and similarly in most of the cases which have been 

 considered above. The imprudent extension of such equivalent 

 forms, which has arisen from the omission of the necessary 

 signs of discontinuity, has frequently led to very erroneous 

 conclusions : thus, the equation 



.r» 1 1 r.r2 - 1 1 r* - 1 1 ^^ - 1 . )t 



-D.« log a: = 2-|^^— I + -J . (^-.qpT)^+ y • (:,qri)a+ &«• \ 



which is true for all values of x between and co , has been 

 extended to all values of x between — 00 and + co , and has 

 thus been made the foundation of an argument for the identity 

 of the logarithms of the same number, both when positive and 

 negative. 



There are two species of discontinuity which we have consi- 

 dered above, one of which may be called instantaneous and the 

 oi\\ev finite : the first generally accompanies such changes of 

 form as are consequent upon the introduction of critical values 



* These formulae would require generally a correction at their limits, in 

 order to render them symbolically general. The nature of these corrections 

 may in most cases be easily applied from the observations which we have 

 made above. 



t This series is given by M. Bouvier in the 14th volume of Gcrgoniie's 

 Aimalcs ties Malhemutifjiies. The conclusion referred to in the text assumes 

 the identity of the logarithms of x" and of ( — .r)^, which is in fact the wholo 

 (jucstion in dispute. 



