911 



alread}' obtained forming with the latter a more or less complete system. 

 The development and (he diseu&sion of all these formulae is the 

 subject of the following pages. 



30. In all symbolic formulae to be treated of, the closed expressions 

 occurring in the right-hand members of them must be developed accord- 

 ing to ascending powers of one or more letters, these powers 

 having no meaning in themselves, but obtaining one when the 

 exponents (upper indices) are replaced by (lower) indices. Now, 

 often certain reductions are allowed which would also be valid if 

 the letters denoted variable quantities, whether or not being restricted 

 to certain domains. 'ë)nQ\\Ye<\nQ,{\or\s we %\\&]\ (taW analytical reductions. 

 The principal condition which should be noticed in order to be 

 able to perform an analytical reduction with symbolic expressions 

 is that equal symbols occurring in different parts of them, have 

 the same meaning, this being the same fundamental condition if (he 

 letters denote numbers. 



Generally speaking an analytic reduction is permitted if the /^^'O^^e/' 

 meaning of the result is the result of the proper meanings, when 

 by the latter the result is meant which would be obtained if the 

 proper meanings were introduced before tlie reduction mentioned 

 is performed. Thus we may have an analijtic sum of .symbols or 

 a product. In the first case we shall often have to apply the rule 

 that in a polynomial consisting of symbolic powers of the same 

 letter, before substituting indices for exponents, terms mxoWmg equal 

 powers a^ may be added analytically. For the proper meanings of 

 such terms are quantities involving equal factors a^-, the coefticients 

 in the symbolic terms being respectively equal to those in the proper 

 ones. The sum of these coefticients multiplied by a* is the analytic 

 sum of (he symbols and the same sum multiplied by au is the sum 

 of (he corresponding proper expressions. Thus the latter sum is 

 indeed the proper meaning of the former. 



If we have a product of symbolic powers of the same letter a, 

 we should carefully state whether the product of their proper mean- 

 ings is meant by it, or the proper meaning of the analytic product, 

 that is of the single power which is obtained by multiplying the 

 powers of a according to the ordinary rule giving as new exponent 

 the sum of the partial exponents. For the proper meaning of the 

 analytic product of a certain number of powers of tlie same letter 

 a is not in general equal to the product of proper meanings of all 

 factors. ') We shall always have to deal with such products of 



1) See, however, the example in N*^. 35. 



64* 



