182 



If the base curve be a hyperbola, whose centre is the 

 origin, the arcs of all the curves of the negative series will 

 depend only on elliptic functions of the first and second kinds. 

 But the general expression for the arc in the positive series 

 contains a function of the third kind, the parameter of which 

 is alternately circular and logarithmic: the curves of an odd 

 order involving the same function of the circular kind, and 

 those of an even order the same of the logarithmic kind, if 

 the real axis of the base-hyperbola be greater than the ima- 

 ginary, and vice versa. 



Mr. Roberts also shows, that besides the case of the 

 equilateral hyperbola, in which the first positive curve is the 

 lemniscate of Bernouilli, and which has been the only one 

 hitherto noticed, at least as far as he is aware, there are two 

 others, in which the arc of the first positive curve can be ex- 

 pressed by a function of the first kind, with the addition of 

 a circular arc in one case, and of a logarithm in the other. 

 The first of these occurs when the imaginary semiaxis is 



■/S— 1 

 equal to — - — (the distance between the centre and focus 



being unity), and this fraction is the modulus of the function. 

 The other case is furnished by the conjugate hyperbola, and 

 the modulus is complementary. In both these cases func- 

 tions of the third kind disappear from the arcs of the positive 

 curves. 



If the hyperbola be equilateral, and its semiaxis be sup- 

 posed equal to unity, the general equation of the derived 

 curves of both series may be presented under the form 





The successive curves represented by this equation are very 

 curiously related to each other. The following property ap- 

 pears worthy of remark : 



Let Pn-i, Pn, Pn + i bc Corresponding points on the 



