256 ON THE RECENT PROGRESS OF ANALYSIS. 



@. The comparison of elliptic integrals of the same form 

 differing only in the value of the variable, or as it is often 

 called, the amplitude of each. This part of the subject divides 

 1 tself into three heads, corresponding to the three classes of 

 integrals. The fundamental results are to be found in the 

 memoirs of Euler, of which we have already spoken. By Le- 

 gendre however they were more fully developed. 



It is interesting to observe that Legendre suggested that the 

 discovery of Euler (namely that the differential equation 



=Q 



admits an algebraical integral, f(x) being the polynomial 



a + fa + 7# 2 + &C 3 + ex*) 



might be generalised, if we consider the differential equation 

 dx + _<fy_ + { dz ==Q 



^/(*) #(y) V/w 



He remarks that this is perhaps the only way in which it can be 

 generalised. 



7. Theorems relating to the comparison of different kinds 

 of elliptic functions. One of the most remarkable of these is 

 the relation between the complete integrals (those, namely, in 

 which the variable x is unity) of the first and second kind, the 

 moduli of which are complementary; that is, the sum of the 

 squares of whose moduli is equal to unity. Legendre's demonstra- 

 tion of it is rather indirect, but many others have been since 

 given. Another theorem may be mentioned, that the complete 

 integral of the third kind can always be expressed by means 

 of the complete integrals of the first and second. A third and 

 most important result shows that in elliptic integrals of the 

 third kind we may distinguish two separate species, and that 

 to one or other of these any such integral may be reduced. A 

 memorable discovery of M. Jacobi has greatly increased the 

 importance of this subdivision, of which we shall hereafter 

 speak more fully. This part of the subject is, I imagine, 

 entirely due to Legendre. 



8. The evaluation of elliptic integrals by means of ex- 

 pansions. 



e. The method of successive transformations. The idea of 



