326 EDWARD SANG ON THE APPROXIMATION TO THE ROOTS, ETC. 



-| q 1 9>r o 09J. 



If we select here the last term of each group as ^ , g- , -^ , -^ , &c, or the 



2 37 590 

 first term as y, ^r, ^» &c, we form a progression with the multipliers q = 1 , 



^? = 16. Similarly, for the square root of 20, we get the quotients 4; 2, 8; 



2,8; &c, occurring in groups of two, the successive approximations being 



4 2 8 2 8 2 



1.4 9.76 161.1364 2 889 & in which the terms - £ ^ £889 &c 

 ' T' 2 ' 17 ' "36 ' "305 ' "64T ' ^ C '' m W ° S ' 2 ' 36 ' 646 ' <BC '' 



progress with the multipliers q = — 1 , _p = 1 8 . 



This circumstance greatly facilitates our investigations in quadratics ; thus 



if the indeterminate equation 



X*=7y*+1 



were proposed, we have at once the solution by seeking J7, and taking the 

 last term of each group : thus 



x— 8, y= 3, 



x= 127, y= 48, 



«=2024 , ?/=rl765 , and so on. 



are the solutions. 



When the group of quotients consists of two terms a, 6, the order of 

 recurrence is given by q— — 1 , p = a8 + 2. 



For a period with the three quotients a, (3 , y , we have q= + 1 , p = a fiy 

 + a + /3 + y. 



For one with the four, a , ft , y , 8 , we have q = — 1 , p — afiyS + (a + y) 

 08 + S) + 2. 



The subject, however, is too extensive to be treated as an appendix to the 

 present paper. 



