REPORT ON CERTAIN BRANCHES OF ANALYSIS. 207 



nence of equivalent forms, which is our proper guide in the 

 establishment of the fundamental propositions of symbolical 

 algebra, in the invention of the requisite signs, and in the de- 

 termination of their symbolical form : but in the absence of the 

 complete enunciation of that law, we may proceed with the in- 

 vestigation of the fundamental rules for addition, subtraction, 

 multiplication and division, and of the theorems for the collec- 

 tion of multiples, and for the multiplication and involution of 

 powers of the same symbol, which will, in fact, form a series of 

 assumptions which are not arbitrary, but subordinate to the 

 conditions which are imposed by our hypotheses : but if we 

 suppose those conditions to be incorporated into one general 

 law, whose truth and universality are admitted, then those as- 

 sumptions become necessary consequences of this law, and 

 must be considered in the same light with other propositions 

 which follow, directly or indirectly, from the first principles of 

 a demonstrative science. In the same manner, if we assume the 

 existence of such signs as are requisite to secure the universality 

 of the operations, the symbolical form of those signs, and the 

 laws which regulate their use, will be determined by the same 

 principles upon which the ordinary results of symbolical al- 

 gebra are founded. 



The natural and necessary dependence of these two methods 

 of proceeding upon each other being once established, we may 

 adopt either one or the other, as may best suit the form of the 

 investigation which is under consideration : the great and im- 

 portant conclusion to which we arrive in both cases being, the 

 transfer of all the conclusions of arithmetical algebra which are 

 general in form (that is, which do not involve in their expres- 

 sion some restriction which limits the symbols to discontinuous 

 values,) to symbolical algebra, accompanied by the invention or 

 use of such signs (with determinate symbolical forms) as may 

 be necessary to satisfy so general an hypothesis. 



There are many expressions which involve symbols which 

 are necessarily discontinuous in their value, either from the 

 form in which they present themselves in such expressions or 

 from some very obvious conventions in their use : thus, when 

 we say that 



cos x = COS {2r i: -\- x), 

 and — COS x = cos { (2 r -|- 1) tt 4 x} 



propositions which are only true when r is a whole number, 

 the limitation is conveyed (though imperfectly) by the con- 

 ventional use of 2 r and 2 r + 1 to express even and odd num- 

 bers ; for otherwise there would be no sufficient reason for not 



