ON THE PELLIAN EQUATION. 



75 



one, not +1, but — l. The final numbers 1118, 85 are consequently 

 entered not in Table I., but in Table II., viz., the entry in this table is 



173 



85 

 1118 



and thence we calculate the numbers y, x of Table I., viz., these are 



2499849 = 2.(1118)= + ! 

 190060=2.1118.85 



Generally Table II. gives for each value of a, comprised therein, 

 values of a-, y, such that 2/^=a«^—l, and then writing yi=2y^ + l, Xi = 2xy, 

 we have 



= (^2ax- — ly = ia''x* — iax" + 1 = a . 4a;^(aa;-— 1) + 1 = aa;,' + 1 



Vi 



SO that .T,, yi are for the same value of a the values of x, y in Table I, 



It is to be remarked that the heading of Table II. is not perfectly accu- 

 rate, for it purports to give for every value of a, for which a solution 

 exists, a solution of the equation y'^=ax^—l. What it really gives is the 

 solution for each value of a for which the period has a double middle 

 term. But if a.=:a^ + l, then obviously we have a solution 7/ = a, x=ly 

 and for any such value of a the period has a single middle term, viz., the 

 entry in Table I. is 



and we in fact have 



that is 



l--(a=+l)0= =+1 



a=-(a=+ 1)1- =-1 



(2a-+l)--(a- + l) (2a)2= +1 



The foregoing instances of the calculation of x, y in the case of the 

 numbers 209 and 173 suggest a table which may be regarded as an ex- 

 tended form of Degen's tables ; viz., such a table, from a=2 to a^99, is 

 as follows : 



Specimen of extended form of Table in regard to the Pellian Equation. 



