ON THE PELLIAN EQUATION. 81 



The meaning hardly requires explanation ; for each number a we 

 have a series of pairs of increasing numbers, y, x, satisfying a series of 

 equations y^^ax^dib ; thus 



The following table, calculated under the superintendence of the 

 Committee, extends froma=1001 to a^lSOO (square numbers omitted) ; 

 it is (with slight typographical variations) nearly but not exactly in the 

 form of Degen's Table I., the chief difference being that for a number a 

 having a double middle term, or of the form a^ + \ (such number being 

 further distinguished by an asterisk), the x, y entered in the table are the 

 solutions, not of the equation y'^:=.aa'!^-\-\, but of the equation y-:=ax^ — \. 

 As T-emarked above, if we have y'^=ax^ — \, then writing 2/i=2i/2 + l and 

 a!,=2a;2/, we obtain y^^^axi^ + l. 



Moreover, for each value of a, in the first line, the first term, which is 

 the integer part of v'a, is separated from the other by a semicolon, 

 and the 1, which is the corresponding first term of the second line, is 

 omitted. 



The calculations were made by C. B. Bickmore, M.A., of New 

 College, Oxford : his values for x and y have been revised as presently 

 mentioned, but it has been assumed that his values for the periods and 

 subsidiary numbers (forming the first and second lines of each division of 

 the table) are accurate ; in fact, any error therein would cause the resulting 

 values of x and y to be wildly erroneous ; but (except in a single instance 

 which was accounted for) the errors in x and y were in every case in a 

 single figure or two or three figures only. 



The values of x and y were in every case examined by substitution in 

 the equation {y'^^=ax' + l, or y'^=ax'^ — '\., as the case may be) which 

 should be satisfied by them. These verifications were for the most part 

 made by A. Graham, M.A., of the Observatory, Cambridge. As already 

 mentioned, some errors were detected, and these have been, of course, 

 corrected. The values of x, y given in the table thus satisfy in every 

 case the proper equation i/2=aa!^ + l, or y-=^ax'^ — \ ; on the ground above 

 referred to it is believed that the periods and subsidiary numbers are 

 also accurate. 



It may be remarked, in regard to the verification of the equation 

 y~=.a:i'r-+\ for large values of x and y, it is in practice easier and safer to 

 calculate ax'^-±\, and then to compare the square root thereof with the 

 given valne of y, than to further calculate the value of y"^. 



1893. 



