310 ON THE RECENT PROGRESS OF ANALYSIS. 



Thus these integrals would belong to the first or second kind, 

 according to the value of the index e, and of X the degree of (f>x. 

 But in reality, though the integrals in question are of the first 

 kind (that is, they admit of summation without introducing 

 either an algebraical or logarithmic function) if e be less than 

 a certain limit, yet if it be not so their formula of summation 

 will in general involve both algebraical and logarithmic func- 

 tions. Either may, under certain conditions as to the form of 

 (j)x, disappear, but while <j>x is merely known as the polynomial 

 of the A,th degree, we cannot decide whether the integral is to be 

 referred to the second or third kind. 



I may mention here a very elegant result due to M. Jacobi. 

 It appears in the thirtieth volume of Crelle's Journal, p. 121, and 

 is a generalisation of the fundamental formula for the addition of 

 elliptic arcs. With a slight modification it may be thus stated. 

 If <f)x involve only even powers of x, the highest being a? 4wl , then 



C x x* m dx 

 the sum of the integrals I -j= '= is equal to the product of their 



arguments, that is of the different quantities denoted by the sym- 

 bol x. In this case then the logarithmic function disappears, 

 and the^'integral belongs to the second kind. 



In the twenty-ninth volume of Crelle's Journal there is a 

 paper by M. Eichelot on a question connected with hyper-elliptic 

 integrals. The reader will find in it a good many fully-de- 

 veloped results, which may be considered as particular cases of 

 Abel's theorem. They illustrate the learned author's criticism 

 of Legendre's classification of hyper-elliptic integrals, though 

 they are not adduced for that purpose. 



The function M (vide ante, p. 244) is a function of the arbi- 

 trary quantities a, Z>, ... c, which, as has been remarked, may 

 themselves be considered functions of the arguments a? l5 x 2 , ... x^. 

 To determine M as a function of the last-written quantities is a 

 necessary ulterior step in almost any special application of Abel's 

 theorem, and this M. Eichelot has done in several interesting 

 cases, establishing at the same time the relations which exist 

 among the quantities in question. His investigations, however, 

 have an ulterior purpose, and are not to be considered merely as 

 corollaries from Abel's theorem. 



Another paper of M. Eichelot, on the subject of the Abelian 

 integrals, is found in the sixteenth volume of Crelle's Journal, 



