ON THE RECENT PROGRESS OF ANALYSIS. 2G5 



of the theory more frequently occasion to consider the value of 

 the amplitude as determined by the corresponding value of 

 the integral than vice versa ; and it therefore becomes expedient 

 to frame a notation by which the amplitude may be expressed as 

 a function of the integral. In a paper in the ninth volume of 

 Crelle's Journal by M. Jacobi, which, like many of his writings, 

 contains in a short compass a philosophical view of a wide 

 subject, he has made use of the analogy between circular and 

 elliptic functions to illustrate the importance of the new notation 

 for the latter. When the rnodulus of an elliptic integral of the 

 first kind is equal to zero, the integral becomes 



dx 



which, as we know, is equal to the arc whose sine is x, or to 

 sin" 1 a;. Now this is a function which we have much less often 

 occasion to express than its inverse sin x, and we accordingly 

 always look on the latter as a direct, and on the former as an 

 inverse function. Yet in the case of elliptic functions, the func- 

 tional dependence for which we had an explicit and recognised 

 notation, viz. that of the integral on the amplitude, corresponds 

 to that which in circular functions lias always and almost neces- 

 sarily been treated merely as an inverse function. The origin of 

 this discrepancy is obvious; our knowledge of the nature of 

 circular functions is not derived from the algebraical integrals 

 connected with them, and therefore these integrals are not 

 brought so much into view as in the theory of elliptic functions 

 the corresponding integrals necessarily are ; but it is certain that 

 while the discrepancy continued to exist the subject could never 

 be fully or satisfactorily developed. The maxim " verba vestigia 

 mentis " is as true of mathematical symbols as of the elements of 

 ordinary language. 



We shall see hereafter that Abel took the same step in his 

 first essay on elliptic functions. At present I shall only re- 

 mark, that one of the earliest consequences of the new notation 

 was the recognition of a most important principle, viz. that the 

 ' inverse function' sin am u, that is, the function corresponding 

 to sin u in circular functions, is doubly periodic, or that it 

 retains the same value when u increases by any multiple either 

 of a certain real or of a certain imaginary quantity. Now 



