264 ON THE RECENT PROGRESS OF ANALYSIS. 



where S, like E, is integral. By differentiating and reducing, 

 we then show that 



dy - - j^ 2 dx, 



v I 



and combining these two results obtain the required verification. 

 The essence of M. Jacobi's demonstration consists in showing 

 that if the value of y in terms of x is such that an equation of the 

 form (a) subsists, then necessarily 



^ = (8} 



where /z- is a constant ; the existence of the two equations (a) and 

 (/3) being equivalent to the two conditions of which we have 

 already spoken (p. 259). In the particular case we are now con- 

 sidering, 



fJL= - . 



v 



15. After considering the transformation of the fifth order 

 (in which the modular equation is 



u" - v 6 + 5wV {u* - v 2 } + uv [I - wVJ = 0), 



M. Jacobi prepares the way for a more general investigation 

 by introducing a new notation. This step is one of the highest 

 importance. We have been in the habit of calling < the am- 

 plitude of the integral 



d<f> 



let this integral be called u. The new notation is contained 

 in the equation (j> = amw ; or if we call sin <, x, so that 



dx 



-i: 



then x = sin am u. 



A new notation is in itself merely a matter of convenience : 

 what gives it importance is its symbolizing a new mode of con- 

 sidering any subject. We had hitherto been accustomed to look 

 on the value of the elliptic integral as a function of its amplitude, 

 to make the amplitude (if the expression may so be used) the in- 

 dependent variable. But in reality a contrary course is on many 

 accounts to be preferred. We have in the more advanced part 



