74 



EEPOET — 1893. 



the first column), and the third column with o, l, we calculate the num- 

 bers of the second column, 29=2.14 + 1, 159=5.29 + 14, 506=3.159 + 29, 

 &c., and the numbers of the third column in like manner, 2=2.1+0, 

 11 = 5.2+1, 35=3.11+2, &c. ; and then writing down as a fourth column 

 the numbers of the second row with the signs +, — alternately, we have 

 a series of equations y^— aa;^=:+A, viz., 



1^-209.0= =+1 



14- -209.1- =-13 

 29- -209.2* = -I- 5 



the last of them being 



(46551)=-209(3220)== + 1 



this last corresponding as above to the value + 1, and the numbers 46551 

 and 3220 being accordingly the y and x given in the fourth and third 

 rows of the table. 



As to the second of the foregoing numbers, 173, the only difference ia 

 that the period has a double middle term, viz., the entry in the Table I. 

 is 



173 



13, 6, ( 1, 1) 

 1, 4, (13, 13) 



190060 

 2499849 



The first row gives the expression of \/173, viz., that is 



11 !1 1 1 



a/173 = 13 + 



6-f (1)+ (l)-H 6-h 26 + 



&c., 



the denominators being 6, 1, 1, 6, then 26 (the double of the integer part 

 13), and then again 6, l, l, 6, and so on. In the second row I remark 

 that Degen prints the parentlaeses (13, 13) for the double middle term. 



The process for the calculation of the x, y is similar to that in the 

 former case, viz., we have 



where the second and third columns begin 1, 13 and 0, l respectively, 

 and the remaining terms are calculated 79 = 6.13 + 1, 92^1.79 + 13, &c., 

 and 6=6.1 + 0, 7=1.6 + 1, &c. ; and then writing down as a fourth column 

 the terms of the second row with the signs + , — alternately, we have 



the last equation being 



(1118)=-173(85)==- 1 



the term for the last equation being always in a case such as the present 



