262 THIRD HEPORT 1833. 



of the variables, when the corresponding equivalent form no 

 longer exists, or when the conditions which determined its exist- 

 ence no longer apply ; the second restricts the existence of the 

 equivalent form to limits of the variable which have a finite dif- 

 ference from each other. In neither case, if we suppose the con- 

 ditions of the discontinuity to be implicitly involved, or if we 

 suppose the explicit signs of discontinuity to be assumed con- 

 formably to the general laws of algebra, can we consider the 

 law of the permanence of equivalent forms to be violated. It is 

 only when a continuous formula is assumed to be equivalent to 

 a discontinuous formula, without the introduction of the requi- 

 site sign of discontinuity to limit the extent of the continuous 

 formula, that we can suppose this fundamental law to be vio- 

 lated or the asserted equation between such expressions to be 

 false. Many important errors have been introduced into ana- 

 lysis from the neglect of those conditions. 



The identity of the values of powers of 1, whose indices 

 are general whole numbers, and also of the sines and cosines 

 of angles which differ from each other by integral multiples of 

 SGO'', is a frequent source of error in the generalization of equi- 

 valent forms, when the symbols which express those indices or 

 multiples are no longer whole numbers. A very remarkable 

 example of both these sources of error has occurred in the for- 

 mula 



(2 cos .r)"' = cos m {2 r it + x) + in cos (»/ — 2) (2 r w + x) 



H \—Q — COS (w — 4) (2r7r -f- x) + &c. 



+ v'^l {*•" '" (2 rir + x) + ni sin {tn — 2) (2 /• * -F- x) 



+ ??i_(j'l^_l) sin {m - 4) (2 r TT -h :r) + &c. (1.) 



If we suppose in to be a whole number, this equation degene- 

 rates into 



/^ s / nx W [tn — 1) 



(2 cos xy" — cos m x + m cos [m — 2) jt -i ^j — ^ — -' 



cos (m — 4) a: -}- &c. (2.) 



the series first discovered by Euler, and which he assumed to 

 be true for all values of in. If, however, we should suppose m 



to be a fraction of the form — ' we should have a values of 



the first member of the equation (2.), and only one of the second. 

 And if we should confine our attention to the arithmetical 

 value fp) of the fust, it would not be equal to the second, un- 



