528 Prof. Sylvester's Notes to the 



fayfrj & • • . to be all linear functions of x } each binomial equation 



out of its entire stock of i roots will contribute — 5 — = roots 



k— 



available towards the solution of the given equation. Mr. Cayley 

 has remarked to me the analogy between this determination and 

 Newton's method of finding the form of the several parabolic 

 equations y — cx^ which represent the branches of a given alge- 

 braical curve at its origin. In the equation to the given curve 

 c# x is to be substituted for y ; the terms will then all become 

 powers of x (an infinitesimal) whose indices will be linear func- 

 tions of X ; every pair of them in turn is equated to zero, and of 

 all the values of \ thus obtained only those will be preserved 

 which cause the two equated linear functions of X belonging to 

 any given pair of terms to be less than all the others, and con- 

 sequently the terms themselves (whose indices the linear func- 

 tions are) infinitely greater than all the other terms. 



Linear functions of a variable figure in both investigations, 

 viz. in Newton's as indices of the same infinitesimal quantity, in 

 mine as quantities whose infinite index is the same* ; but the 

 logic and mode of procedure (utterly unlike as are the questions 

 in their origin and subject matter) is the same in either case. 



Note D. 

 The remark contained in the preceding note, as to the effect 



of representing N * by an infinite rational fraction being iden- 

 tical with that of expressing it as a definite integral, combined 

 with a consideration of the cause of the success of the particular 

 method referred to in Note B, has led me to the investigation 

 following, of the value of the complete elliptic function of the 

 first species. As usual denoting it by F(c), we have 



F( C )=r 



Jo 



\0- 



^l-c^sinfl) 2 



-Sfl/df 00 / cosfl 



~?rJo Jo ^l-c 2 (sin0)* + (cos0)V 



* In a word, Newton's equation is an exponential one made up of 

 nothings, mine an algebraical one made up of infinities. 



