AS COMMONLY INVESTIGATED. 233 



The expression 



@,x 



would, according to this notation, express an abscissa that, regularly increasing 



from to TT,* recommences with the latter value, until it has again increased 



to the same amount; whence it follows that whatever may be the form of F, 



we should have 



'@.Fx + F.'^.x. 



It would be out of place to repeat here the deductions which I have else- 

 where drawn from this and similar equations; and my only object in now no- 

 ticing them is to remark, that were such conventions agreed upon, every spe- 

 cies of discontinuity might be reduced to a class, and rendered a continuity of 

 an inferior order, the connexion with the other arranged functions of the cal- 

 culus being established by seeking expressions for such simple periodic ele- 

 ments as we have here denoted by '© x. 



I shall not insist further on this subject, having said enough to render clear 

 all that I have to urge in regard to the principal topic discussed in this paper. 

 It has appeared to me that much confusion has arisen from the vague notions 

 entertained concerning modal functions ; functions of the calculus, or of nume- 

 rical logic; continuous functions, and discontinuous functions; and having, in 

 the preceding section, illustrated the real distinction between the two former, 

 I trust I have said enough in the present section to show, not only that the dis- 

 tinction between the latter is merely that of a superior and inferior continuity, 

 but also that when we assume with Poisson every function of the form 



{F(X,h)=:0} 

 h:=0 



to contain a factor h", we must, at best, form our conclusion on a very limited 

 class of mathematical expressions. 



Section III. 



Functions arranged in the Order of their Magnitudes.— Tdyhr' s Theorem. 

 Functions X, X, X, &c., of x, that vanish when x = a, are said to be ar- 



1 2 3 



ranged in the order of the evanescent magnitudes which they assume for x = a, 

 when for every value of n we have 



* »»■ is used to denote any period, the symbol ■^ being left for the common period of trigonometri- 

 cal functions. 



VII. — 3 I 



