ON ELLIPTIC AND IIYPERELLIPTIC FUNCTIONS. 113 



The following formulae may be regarded as fundamental ; they may be seen 

 proved in Schellbach, Section 130 : — ■ 



J Jta- /«;-— 1 ^ / - 0,(« + -"^O 



There ii^alao a paper on the third elliptic integral by Professor Somoff in 

 the 47th volume of Crelle's Journal, written to facilitate the numerical cal- 

 culation of its value. There are a few papers on elliptic functions, connected 

 with Abel's theorem, and the multiplication of functions 9, which I hope 

 to consider hereafter, when treating on hyperelliptic functions, with which 

 they are closely related. 



Section 14. — 1 have long wished to see a treatise on elliptic functions 

 written on the following plan. First, I have wished the subject to be con- 

 sidered as if consisting of three parts — evolution, division, and transforma- 

 tion. This, indeed, has been in great part effected in Abel's great memoir 

 on the subject ; but this memoir, it wiU be observed, contains no indication 

 of the existence of the functions 0*. The evolution of elliptic functions 

 should be effected in the following way: — They should be expressed by 

 doubly infinite products, and this should be done by a method closely re- 

 sembling that employed by Abel. These doubly infinite products should 

 then be transformed into the singly infinite products used by Jacobi ; and 

 lastly, these singly infinite products should be multiplied together, so as to 

 form the functions d. The division and transformation should be effected 

 separately, and the evolution deduced as effected by Abel, and not in an 

 elementary treatise, derived from transformation, as we see in the ' Funda- 

 menta jS'ova.' 



An excellent treatise on elliptic functions, which forms a part of Ber- 

 trand's ' Traite de Calcul Differentiel et de Calcul Integral,' now publishing 

 in France, keeps these objects steadily in view, and 1 have great pleasure in 

 recommending it to the reader. If I do not dwell longer on this work, it 

 is not assuredly because I am insensible to its merit, but because it is not 

 only written in the highest style of mathematical elegance, but is also so 

 perspicuous that any commentary from me would be superfluous. 



Pakt IV. 



Section 1. — It will be well to commence this part of our work with show- 

 ing how elliptic functions may be applied to finding the area of the surface 

 of an ellipsoid. 



Let ax^ + hi/^+cz''-=l be the equation to the surface of an ellipsoid, y the 

 angle which the normal makes with the axis of (z), ^ the angle which the 



* He alludes, however, to these functions in his subsequent writings, after the discoveries 

 of Jacobi. 

 1870. . t 



