210 SCIENCE PROGRESS 



it can be divided into two parts separated by the sign of 

 equality, i.e. it may be put in a variety of ways into the form — 



y = M x ) = fa(*) 



If the graphs of fi{x) and f 2 {x) are obtained, the points of 

 intersection of the two curves are, of course, the values required. 

 Now it may be difficult to obtain the graphs, and even if they 

 are obtained, as the problem is to find the numerical value of 

 the roots, we are just as far as ever from obtaining the value 

 at the point. Thus we might continually approach the value 

 along a certain path without ever reaching it, or we might 

 oscillate about it. 



We have no room here to enter into the history of solution 

 by approximation, but some of the difficulties in the solution 

 by the method of iteration will be seen from the consideration 

 of the following examples. If we take the cubic equation — 



x* — 15*" + 75a? — 125 =0 



which has three roots each equal to 5, as the coefficients are 

 symmetric functions of the roots, the last term 125 being the 

 product of those roots, we should expect a small change in 

 the coefficients to produce correspondingly small changes in 

 the roots, but by changing 125 to 124 we have the equation — 



x l — i$x* + 75* — 124 = o 



whose roots are 4 and — — (two imaginaries). Now 



this curious property of equations has been long known. 

 James Logan, born at Lurgan, Armagh, on October 20, 1674, 

 who accompanied William Penn, as Secretary, to America 

 in 1699, in a letter to Sir Wm. Jones, dated from Philadelphia, 

 May 4, 1738, gives the following equation : 



x* — 24X 1 + 2i6x* — S64X + 1296 = o 



which has four equal roots 6, and he observes : If you change 

 the — 24^ to — 25^ the root will sink from 6 to less than 

 3-5, and if you change it to — 23** the roots are imaginary. 

 Or we may consider the equation — 



x* — 3x + 2' 00000 1 = o. 



One of the roots being obviously very nearly equal to 1 , if we 



