246 ON THE RECENT PROGRESS OF ANALYSTS. 



Hence, as ^x = sm~ 1 x, our result is merely this, that the 

 sum of two arcs is constant if the sine of one is equal to the 

 cosine of the other. 



An infinity of analogous results may be obtained either by 

 varying the form of y (e.g. by making y = \ll-x*), or by 

 changing the equation (1). A formula applicable to all forms 

 of y, and which, for each, includes all the results which can be 

 established with respect to it, is, it will readily be acknow- 

 ledged, one of the most general in the whole range of analysis. 

 Abel's principal result is a formula of this nature ; he developed 

 at considerable length the various consequences which may be 

 deduced from it. 



Generally speaking, the number of independent variables 

 a, Z>, . . . c will be less than that of the different roots, x l . . . x^ ; 

 hence a certain number, say m, of the roots may be looked on as 

 independent (viz. as many as there are quantities a, b, ... c), and 

 the rest will be functions of these. It may be shown that it will 

 always be possible to make the difference p m constant, so 

 that the sum of any number of the transcendents *fy is ex- 

 pressible by a fixed number of them, together with an algebraical 

 and logarithmic function of the arguments, i. e. of x l , . . . x m . 

 In the case of elliptic integrals, it had long been known that the 

 sum of two may be thus expressed by a third ; and Legendre 

 pointed out that the sum of any number may similarly be ex- 

 pressed by means of one. Accordingly it appears from the general 

 theory, that in this case fju m may be made equal to unity. 



9. The history of this important theory is curious. It was 

 developed by Abel in an essay which he presented to the In- 

 stitute in the autumn of 1826, when he had scarcely completed 

 his twenty-fourth year. 



In a letter to M. Holmboe, appended to the edition of his 

 collected works, Abel writes, ' Je viens de finir un grand traite 

 sur une certaine classe de fonctions transcendantes pour le pre- 

 senter a 1'Institut, ce .qui aura lieu lundi prochain. J'ose dire 

 sans ostentation que c'est un traite' dont on sera satisfait. Je 

 suis curieux d'entendre Topinion de 1'Institut la dessus. Je ne 

 manquerai pas de t'en faire part.' Long before this memoir was 

 published, Abel had become * chill to praise or blame.' He 

 died at Christiania in the spring of 1829. 



