392 Mr Murphy on the Inverse Method of 



Now it is evident that 



= (l-2t).^Atty- l -2An-l).~.(tt'r'; 



.-. P.=(l-a/).P»_ I -3<«-rl).jEa? B _ I .' 



Suppose now that for a particular value of n, all the roots 

 of the equation P„-, = are real (as t„ t>, t 3t . ..#„_,); they lie be- 

 tween the roots of the equation /P„_, = 0, and from the nature 

 of the question it is obvious that they all lie between and 1 ; 

 substitute them in the order of their magnitude for t in the above 

 equation, then since P„_, vanishes, the resulting- values of P, will 

 have alternately opposite signs. Again, when t = 0, P„ = 1=P„.,; 

 and therefore P„_,, and consequently its integral remains positive 

 from £ = to t=.t x the least root: the order of the signs of fiP„-i is 



therefore +, -, +, -, the last in the series being + or - 



according as n is even or odd, and the corresponding signs of 

 P„ are -, + , -, + &c. in number (» — 1) also. If we put t = 0, 

 P, is then =1, and therefore its sign is + ; and if we put t=\ 

 then P, = ( - 1)" ; it follows therefore that the substitution of 

 0, A, ti,...t n „i, 1, in P for t gives n alternations of signs, therefore 

 the roots of the equation P„ = are all real on the supposition 

 that those of P_i=0 are such. 



Now by Art. 29, there is one transition line when » = 1, hence 

 there are two such lines when w = 2, 3 when w = 3 &c. so that 

 generally there are n such lines symmetrically situated with re- 

 spect to the points A and B, (vid. Fig. 5, where the black lines 

 represent the transition circles, which divide the surface of the 



