705 



to elliptic integrals as already established, which shows that he was 

 not seeking for a new basis of argument, but only for new proper- 

 ties. The author of the present paper proposes (for didactic pur- 

 poses) to commence the higher theory from these functions. The 

 first division of his essay is purely algebraic and trigonometrical, not 

 introducing the idea of elliptic integrals at all. Adopting as the defi- 

 nition of the functions A and the two equations 



A(q, ,z > ) = 2<^(sin,r q*'' 2 sin3x + q 2 - 3 sin 5x &c.) 



6(q, x)=l2q 1 ' 1 cos 2x + 2q 2 - 2 cos 4r 2^ 3>3 cos 6# + &c. 



it demonstrates by direct algebraic methods many properties of great 

 generality, of which we shall here specify 



1. If Vb stands for 6 / g> T ), and Vc for A fo^, which is 



" &i*0 



shown to yield 6 2 + c 2 = 1 ; and if, further, A stand for A(q, x + JTT), 

 Q(q, X + %TT); we get the four equations (equivalent to two only) 



A 2 + A 02 =c0 2 ; A 02 + A 2 =c0 02 ; 



from which it directly follows, that if w is an arc defined by the 



equation Vb tan w= ^, we shall have simultaneously Vc sin w= ; 

 A 



Vc cos w= Vb ; A/(! -c 2 sin 2 o>)= Vb . The symbol A(c, a,), 

 



A(w) or A represents V(l c 2 sin 2 w) in this theory. 

 2. It is further shown that 



dx 

 whence is easily obtained 



dx dx dx 



3. By direct multiplication of two trigonometrical series, it is 

 found that 



)A(?, y)= A( ? 2 , x-y).Q(q\ x+y) 



\ x+y) 

 ,^ + y). 

 From the property marked (2) we obtain the connexion of the func- 



