ON THE RECENT PROGRESS OF ANALYSIS. 263 



but also sufficient, or, in other words, that it involves the 

 second condition already mentioned. 



The transcendent I -7= . 8 may be transformed 



by assuming y y, U being composed wholly of odd powers 



of x, and F of even powers of it. If the degree of U be greater 

 than that of F, the transformation is said to be of an odd order, 

 and of an even order in the contrary case. 



This being premised, M. Jacobi discusses the particular 

 cases of the transformations of the third and of the fifth order. 

 The first is the same as that of Legeridre. It is shown that if 

 we put 



(v + 2u 3 ] vx -f u?x 3 

 y ~ v* + V*K? (v + 2u s ) x* ' 



where u and v are constants connected by the following equa- 

 tion 



we shall get 



__ _ dy _ v + 2u? dx 



in which k = u* and \ = v 4 . The equation connecting u and v 

 is called the modular equation. 



The ' principle of double substitution' may be illustrated by 



writing -- for a? in the expression for y, which, then becomes, 

 u x 



according to the principle in question, - . 



If we seek to show that the assigned value of y actually 

 satisfies the differential equation just stated, we begin by find- 

 ing the value of 1 y. Reducing this value by means of the 



7? 2 

 equation between u and v, we can put it in the form (1 x) -y , 



R being an integral function of x and F, as heretofore the 

 denominator of the expression for y. The value of 1 + y is hence 

 got by changing the sign of x and y, while that of 1 v*y is ob- 



tained by simultaneously replacing x and y respectively by -4- 

 .- u x 



and and reducing. Similarly for 1 + v*y. Hence it will 



c/ 



appear that 



