1904.] LAMBEKT — EXPANSIONS OF ALGEBRAIC FUNCTIONS. 167 



The limit for the terms 2, 6, 7 is infinite, the limit for terms 6, 7, 

 8 is zero, the limit for the terms 2, 7, 8 is infinite. Hence term 6 

 permanently underscored, term 7 is not underscored, and term 8 is 

 temporarily underscored. 



The limit for terms 6, 8, 9 is infinite, for terms 8, 9, 10 zero, 

 for terms 6, 8, 10 infinite. Hence term 8 is permanently under- 

 scored, term 9 is not underscored, and term 10 is temporarily 

 underscored. 



The limit for terms 8, 10, 11 is infinite, hence term 10 is 

 permanently underscored. 



The limit for terms 10, 11, 12 is finite, and these three terms 

 are underscored as one term. 



The several equations formed by retaining in equation (i) in 

 succession only consecutive underscored terms, if these terms are 

 single, and if a group of terms is underscored by retaining only the 

 group of terms, the first term of the group and the next preceding 

 underscored term, and the last term of the group and the next 

 succeeding underscored term, will determine the first approxima- 

 tions of the fourteen branches of the function. 



These equations are 



a) Gx'y -j- F= o, b) Fx'/ -{-E=^o, c) Ef + Dx^ = o, 

 d) JDy' + Cx' = o, e) Cy" + Bx'y -f- Ax' = o, 



and the fourteen first approximations are 



Of these fourteen branches the separate branch a) and the three 

 separate branches b) go through infinity when x = o. The cycle 

 of five branches c), the cycle of three branches d), and the two 

 separate branches e) constitute the ten branches which meet at the 

 singular point. 



If a factor t is introduced in succession into all the terms of 

 equation (i) except the terms used to determine the first approxi- 

 mation of a branch of the function, the successive approximations of 

 this branch are determined by developing _>' in ascending powers of 

 /, X considered constant, by Maclaurin's Series and making / unity 

 in the result. 



