706 

 tions A, with elliptic integrals. For, if F(c, w), as usual, stands for 



f _ * 



JXO-c 2 sm 2 u,)' 

 it yields 



* "XX. ;$$= 



This introduces the second and principal part of the essay. An easy 

 inference from (3) is, that 



and consequently that if rj is related to (f and to x -f y by the same 

 law as w is to q and to x, while c / is to q 2 what c is to q, we obtain 



4. - 



when F(e,,,)_F(c. M 



- 



This formula has the peculiarity of comprising Euler's integrals, 

 with the integrations of Lagrange and of Gauss; namely, if w=0, 

 we get the scale of Lagrange. If 0=0, the scale of Gauss is ob- 

 tained. But if we introduce a new variable , such that 



we eliminate rj by aid of the last result 



and obtain 



which is equivalent to Euler's integration. 



The author believes this generalization to be new. 



5. He proceeds (assuming now the theory of Lagrange' s scale) to 

 prove the higher theorems by much simpler processes. E being the 



second elliptic integral, he writes G for E ~ F, and V for J Gc?F, 



and out of the integration Jlog A=|V / V (where V, is to (f and 

 2x, what V is to q and x), he deduces 



