216 Methods for the Solution of Special Problems [OH. vm 



250. This last formula supplies the easiest way of calculating actual 

 values of 1^. The values of 7J, P 2 , ... 7? are found to be 



P 9 (n) = T V (23V - 315/i 4 + 105/. 2 - 5), 

 P 7 fa) = ^ (429// - 693/a' + 315//, 3 - 35/4 



251. The equation (/J? l) n = has 2?i real roots, of which n may be 

 regarded as coinciding at p. = 1, and n at //, = 1. By a well-known theorem, 

 the first derived equation, 



will have 2n 1 real roots separating those of the original equation. 

 Passing to the nih derived equation, we find that the equation 



has n real roots, and that these must all lie between /Lt = -l and /Lt = + l. 

 The roots are all separate, for two roots could only be coincident if the 

 original equation (^-1)^ = had n + 1 coincident roots. 



Thus the n roots of the equation P n (AI) = O are all real and separate and 

 lie between /A = 1 and //, = -f 1. 



252. Putting //, = !, we obtain 



A/1 - 2h + h- 



so that PI = > = . . . = 1. Similarly, when p = - 1, we find (cf. 240) that 



-= + 7>=-^=... = -l. 



We can now shew that throughout the range from //, = 1 to p = 

 the numerical value of P n is never greater than unity. We have 



x l + fo- + ^ 



