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 ex- 

 pUcitly, and to construct formulae 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 formulee themselves. We shall now 

 proceed to notice some examples of such formulas. 

 The well known series f 



X 1 1 1 



r TT + -^= sin j; — -^sinSx + -jT-sinox — -T-sin4a: + &c. (1.) 



Z /i O 'if 



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

 and to such values of r ir + — as are included between -3- and 

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



* If a definite integral (C) has n determinate values «„ etj, . . . «», within 

 the limits of the variable a and h, 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 - te.) ( C - gj . . ■ (C - «n) ^ 1: 



* «1 X «2 X . . . «» 



thus in the case considered in the text, we get 



'Dj" = - 2 (C - 1) ("C - ^) + 1 = - 2 C2 + 3 C. 



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

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



= log e =.x a/— 1 + 2 r ^r V— 1 = Ve — e / 



and, therefore, dividing by 2 a/— 1, and replacing the exponential expressions 

 by their equivalent values, we get 



X 111 



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



2 2 3 4 



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



■+ 00 and — <» . 



