458 Cambridge Philosophical Society. 



tion. And the general solution of (1) is found by assuming b and 

 c in terms of «, and then making a a self- compensating variable. 

 24. The paper is concluded by some remarks on notation. 



In an appendix to the preceding paper, read to the Society on the 

 1st of May, 1854, Mr. De Morgan points out an error committed 

 by M. Cauchy in a very remarkable theorem, of which his enuncia- 

 tion is as follows. 



Let <^.i' be a function which can be developed in integer powers 

 oi X. Let r (cos y + sill . V — \),r being positive, be any one of 

 the roots of (^.r= 00 or of 0'^.'= oo . Then the development of 0a: is 

 convergent from a'=0 up to ;i; = the least value of r. 



^L Cauchy stipulates that the function shall be continuous; but 

 he defines a function to be continuous so long as it remains finite, 

 and receives only infinitely small increments from infinitely small 

 accessions to the variable. It is then obviously impossible that the 

 above theorem should be universally true. Were it so, it would 

 follow that the development of (l+.r)- is convergent for all finite 

 values of x, whereas it is well known that this development becomes 

 divergent when x is greater than unity. The error of M. Cauchy's 

 demonstration (which contains a valuable method for establishing a 

 large class of definite integrals) is the assumption that if an infinite 

 number of convergent series of the form a-^bx-^cx- + . . . , all with 

 one value of x but different values of a, 6, c, .... be added together, 

 the sum divided by the number of series is also a convergent series. 

 This assumption is not universally true. 



Mr. De Morgan takes a totally different line of demonstration, 

 and establishes the following theorems. 



If ?-(cos -h sin . -/—!), r being positive, represent a root of 

 any one of the equations 0j-=oo , (p'x=-^^ , 0"a;=co , . . . . then the 

 development of (px in powers of x is always convergent from x-=0 

 up to x= the least value of r, and divergent for all greater values of x. 



If the development have all its coefficients positive, or if all beyond 

 an assignable coefficient be positive, the least value of r is obtained 

 from a real and positive root. 



If the signs of the development be, or finally become, recurring 

 cycles, with / in each cycle, the least value of r is obtained from a 

 root in which cos + sin 9 . a/— 1 is one of the Ith. roots of unity. 

 If no such cycle be finally established, cos 0-|-sin0 . \^ — 1 may have 

 a value of which is incommensurable with the right angle. 



M. Cauchy has established from his own theorem (the want of 

 sufficient statement of conditions not affecting this particular case) 

 the necessity of the observed fact, that the developments produced 

 by Lagrange's theorem for the development of implied functions 

 always give, when convergent, the least of the real values which are 

 implied. 



