ON THE PELLIAN EQUATION. 



73 



Tables connected with the Pellian Equation from the 'point where 

 the work luas left by Degen in 1817. — Report of the Committee, 

 consisting of Professor A. Cayley, Dr. A. E. Forsyth, Professor 

 A. Lodge, and Professor J. J. Sylvester. (Dratvn up by 

 Professor Cayley.) 



We have on the Pellian Equation Degen's tables, the title of which is 

 ' Canon Pellianus sive Tabula simplicissimam fequationis celebratissim£e 

 y^=ax!^ + l solutionem pro singulis numeri dati valoribus ab 1 usque ad 

 1000 in numeris rationalibus iisdemque integris exhibens. Autore Carolo 

 Ferdinando Degen. Hafnise, apud Gerhardum Bonnierum, mdcccxvii. 8°. 

 Introductio, pp. v-xxiv. Tabula I. Solutionem sequationis y^ — aa;^— 1=0 

 exhibens, pp. 3-106. Tabula II. Solutionem sequationis y^ — ax"+l=0, 

 quotiescunque valor ipsius a talem admiserit, exhibens, pp. 109-112.' 



The mode of calculation is explained in the Introduction, and illustrated 

 by the examples of the numbers 209, 173. 



As to the first of these the entry in Table I. is 



where the first line gives the expression of v/209 as a continued frac 

 tion, viz., we have 



a/209 = 14 + 



2 + 



1^ 



5 + 



1 



3 + 



1 



2 + 



1 

 3 + 



1 



5 + 



1^ 



2 + 



1^ 



28 + 



2+ ""'■' 



the denominators being 2, 5, 3, (2), 3, 5, 2, then 28, which is the double 

 of the integer part 14, and then again 2, 5, 3, (2), 3, 5, 2, and so on, the 

 parentheses of the (2) being used to indicate that this is the middle term 

 of the period. 



The second row gives auxiliary numbers occurring in the calculation 

 of the first row and having a meaning, as will presently appear. Observe 

 that the 11 which comes under the (2) should also be printed in paren- 

 theses (11), but this is not done. 



The process for the calculation of the x, y is as follows : 



viz., writing down as a first column the numbers of the first row, and 

 beginning the second column with l, 14 (i4 the number at the head of 



