ON MATHEMATICAL TABLES. 319 



Even here, for some of the values of o, the values of x, y are extremely large ; 

 thus a=61, .1-226,153,980, i/=l ,766,399,049. 



And prohably tables of a like extent may be found elsewhere ; in particular 

 a table of the solution of y-=^a.v' + l (— when the value of a is such that 

 there is a solution of 2/-=f^^'"— 1, and + for other values of a), a=2 to 135 

 is given, Legendre, ' Thcorie des JSTombres,' 2nd ed. 1808, in the Table X. 

 (one page) at the end of the vrork. For the before-mentioned number 61 

 the equation is ?/^=61 cir-\, and the values are .r=3805, 2/=29718; much 

 smaller than Euler's values for the equation 7/'-= 61 x'^-\-l. 



28. The most extensive table, however, is 



Degen, " Canon PeUianus, sive Tabula, simplicissimam equationis celebra- 

 tissimse : y'^=ax--\- 1, solutionem, pro singulis numeri dati valoribus ab 1 usque 

 ad 1000 in numeris rationalibus, iisdemque integris exhibens." Auctore 

 Carolo Ferdinando Degen. Hafn (Copenhagen) apud Gerhardum Bonnarum, 

 1817. 8vo, pp. iv to xxiv and 1 to 112. 



The first table (pp. 3-106) is entitled as " Tabula I. Solutionem Equationis 

 y-—(tx^ — \=Q exhibens." It in fact also gives the expression of Va as a 

 continued fraction ; thus a specimen is 



209 



14 2 5 8 (2) 

 1 13 5 8 11 



3220 

 46551 



Here the first line gives the continued fraction, viz, 



V209_14+-_^g^g^2 + 3 + 5 + 2 + 28 + 2 + &c., 



the period being (2, 5, 3_, 2, 3, 5, 2) indicated by 2, 5, 3 (2). [The number 

 of terms in the period is here odd, but it may be even ; for instance, the 

 period (1, 1, 5, 5, 1, 1) is indicated by 1, 1 (5, 5)]. 



The second line contains auxiliary numbers presenting themselves in the 



process; thus R^=239 we have E=14+-, 



1 ^ l(It + 14) ^Il + 14^ 1 



""R-U 209-14" 13 """/B' 



13 _ 13(R + 12) ^ R + 12 ^ 1 • 

 R-12 209-12'' 5 ^y 



_ 5 _ 5(R+13 _ R+13 _ 1 

 '^~R-13~209-13"-~ 8 ~ "^S' 

 (fee, 



where the second line 1, 13, 5 . . . shows the numerical factors of the third 

 column. The value of this second line as a result is not very obvious. 



The third line gives x, and the fourth line ?/. 



29. The second table, pp. 109-112, is entitled " Tabula II. Solutionem 

 rcquationis »/-— rt.r'' + l=0 quotiescunque valor ipsius a talom admiserat, ex- 

 hibens;" viz. it is remarked that this is only possible (but see wfr(Y)ior those 



