Prof. Nevvmau on the Higher Theory of Elliptic Integrals. 395 

 A(ry, 3c)h.\(i, v)=Hr' ^-y).e((f, x+y) 



+ e(q\x-y).A(cf,x + 7/); 

 Q(q, x)Q\q, y) = A(c/, x-y).A(ff, x + y) 

 + e(r/,^'-2/).e(r/,^ + y). 

 From the pi'operty marked (2) we obtain the connexion of the func- 

 tions A, 9 with elhptic integrals. For, if F(p, w), as usual, stands for 

 i dii) 



Jo^(l-c'siira))' 

 it yields f^y v,.^ F(e, w) x 



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

 inference from (3) is, that 



A{q-,x + ij) _ AxM ' y + AyM'x _ 



e(q-, x + y)~exQ''y+ QijO^x ' 

 and consequently that if ?/ is related to q' and to x + y hj the same 

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



4. Vc. sm ?;= -^ ^^ — ^, \. , 



when F(c,,) F(c, c.) + F(c, 8) 



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

 with the integrations of Lagrange and of Gauss: namely, if _w=0, 

 we get the scale of Lagrange ; if 9=0, the scale of Gauss is ob- 

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



_F(e, O = F(c,w) + F(c,0), 

 we eliminate i] by aid of the last result 



(which is Vc, sin, =^^^^), 



and obtain / 1 — Af _ c sin (u> + 9) . 



V r+A^~ Aw + A9 ' 

 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^Go^F, 



and out of the integration §log A=|V,— V (where V, is to q^ and 

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



V=log.^i^ 



by a process fundamentally that of Legendre, Second Supplement, 

 § 196. This is the equation by which E, and indirectly the third 

 integral n, is linked to the functions AG. 



6. We may fuither point out, as perhaps new, the developments of 

 A6 in the case when q is very near to 1 . Let r be related to b as 



«7 to c; then log -.log -=7r-. If log -=7r«, and x=nu, 

 q r q 



