‘ ; ON THE RECENT PROGRESS OF ANALYSIS. 51 
be | - 
where S, like R, is integral. By differentiating and reducing, we then show 
a _v+2u5§ 
me 2 PI ha x; 
i and combining these two results obtain the required verification. 
_ The essence ot M. Jacobi’s demonstration consists in showing that if the 
_ yalue of y in terms of « is such that an equation of the form («.) subsists, 
_ then necessarily d S 
P Se saga ag i asth alu a nabldnrd akncenouggs inked 
t dx V2 
_ where » is a constant; the existence of the two equations (a.) and ({.) being 
_ equivalent to the two conditions of which we have already spoken (p. 48). 
_ In the particular case we are now considering, 
_v+2u 
rt & aera tr 
15. After considering the transformation of the fifth order (in which the 
modular equation is 
us — v8 + 5 uv? {u2 — v2} +40 {1 — wv} =0), 
, M. Jacobi prepares the way for a more general investigation by introducing 
_ anew notation. This step is one of the highest importance.. We have been 
d 
i in the habit of calling ¢ the amplitude of the fin nM a paaie = = in? 
let this integral be called «. The new notation is contained in the equation 
6 = vee or if we call sin ¢, x, so that w -f va =e my 
_ then z= sin am w. 
_ _ A new notation is in itself merely a matter of convenience: what gives it 
_ importance is its symbolizing a new mode of considering 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 independent variable. But in reality a contrary course is on 
Many accounts to be preferred. We have in the more advanced part of the 
theory more frequently occasion to consider the value of the amplitude as 
_ determined by the corresponding value of the integral than vice versd ; 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 vo- 
lume 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 modulus of an 
elliptic integral of the first kind is equal to zero, the integral becomes 
fe dz 
2 A 2 which, as we know, is equal to the are whose sine is z, or to 
° 
‘sin-'z. Now this is a function which we have much less often occa- 
sion to express than its inverse sin 2, and we accordingly always look on the 
latter as a direct, and on the former as an zzverse function. Yet in the case 
_ Of elliptic functions, the functional dependence for which we had an explicit 
and recognised notation, viz. that of the integral on the amplitude, corre- 
ponds to that which in circular functions has always and almost necessarily 
een treated merely as an inverse function. ‘The origin of this discrepancy 
1s obvious; our knowledge of the nature of circular functions is not derived 
: EQ 
