252 THIRD REPORT — 1833. 



limits and also including the limits, and to zero for all other 

 values *. But the expressions which thence arise, though fur- 

 nishing their results in strict conformity with the laws of sym- 

 bolical combinations, possess no advantage in the business of 

 calculation beyond the conventional and arbitrary signs of dis- 

 continuity which we first adopted for this purpose : but though 

 it is frequently useful and necessary to express such signs ea;- 

 plicitly, and to construct formulfE which may answer any as- 

 signed conditions of discontinuity, yet such conditions will be 

 also very commonly involved implicitly, and their existence and 

 character must be ascertained from an examination of the pro- 

 perties of the discontinuous formulae themselves. We shall now 

 proceed to notice some examples of such formulas. 

 The well known series f 



X 1 1 . 1 . 



r TT + -3- = sin j; — ^ sin 2 j: + -?r sin 3 x — -p sin 4.r + &c. (1.) 



is limited to integral values of r, whether positive or negative, 

 and to such values oirit •\- -^ as are included between -^ and 



— -^ : the value of r, therefore, is not arbitrary but condi- 



* If a definite integral (C) has n determinate values »i, «2, . . . un, within 

 the limits of the variable a and b, and no others, the values at those limits 

 being included, and if C be equal to zero for all values beyond those limits, 

 then we shall find 



«D « = _ (C — eti) ( C — eta) • • • (C — »n) ^ j. 

 oil X oCq X • . ■ cin 



thus in the case considered in the text, we get 





= _ 2 (C — 1) ^C — i-) + 1 = — 2 C2 + 3 C. 



t The principle of the introduction of r w in equation (1.) by which it is ge- 

 neralized, will be sufficiently obvious from the following mode of deducing it : 



= log {i + e'^~' } - log {1 + e-' ^^} 



log 



1 I '*' 



1 +e 



, -X V-l ; ^ . - , ■ ( S'/^ -JfV-A 



= log e =. X a/— I + 2r t V — 1 = Ve — e / 



and, therefore, dividing by 2 V* — 1» and replacing the exponential expressions 



by their equivalent values, we get 



X . 1 1 . 1 . 



r IT + — = sin a; sin 2 a; + — sin 3 a; sin 4 a; + &c., 



2 2 3 4 



where x upon the second side of the equation may have any value between 

 •+ 00 and — 00 . 



