64 MR EDWARD SANG ON THE EXTENSION OF BROUNCKER'S METHOD 



parison of the logarithms of the three prime numbers 5, 3, and 2. These 

 logarithms are incommensurable, and so it is impossible to tune a keyed instru- 

 ment perfectly. The comparison of these three logarithms furnishes a con- 

 venient instance of the application of our method. 



Putting, for shortness' sake, A = log 5, B = log 3, C = log 2, we obtain the 

 following equations : — 



■69897 



00043 



= 



A = 



l.B + 0.C + D 



•47712 



12547 



= 



B = 



l.C + 0.D + E 



•30102 



99956 



= 



C = 



l.D + 0.E + F 



•22184 



87496 



= 



D = 



l.E + 0.F + G 



17609 



12590 



= 



E = 



2.F + 0.G + H 



7918 



12460 



= 



F = 



1.G + l.H + J 



4575 



74905 



= 



G = 



2.H + 0.J + K 



1772 



87669 



= 



H = 



l.J + 0.K + L 



1569 



49885 



= 



J = 



l.K + 2.L + M 



1029 



99566 



= 



K = 



5.L + 0.M+ N 



203 



37784 



= 



L = 



1.M + 5.N + P 



132 



74750 



=z 



M = 



10.N + 0.P + Q 



13 



10644 



= 



N = 



2.P + l.Q + R 



5 



09810 



= 



P 





1 



68303 



= 



Q 





1 



22720 



— 



R j 





and, in order to compute from these the successive approximations, we may 

 write the three sets of coefficients, p, q, r, in three horizontal lines, as in the 

 subjoined scheme: — 



r 



1 1 1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



9 















1 











2 







5 







1 



P 



111 



1 



2 



1 



2 



1 



1 



5 



1 



10 



2 



A 1 1 1 2 3 7 9 28 35 44 318 353 5164 10638 

 B 1 1 1 2 5 6 19 24 30 217 241 3525 7267 



C 1 1 1 3 4 12 15 19 137 152 2224 4585 



Having written unit beneath the first p, to serve as the beginning of the 

 series A, we multiply that unit by p x to get 1, the second value of A 2 , which 

 value we write beneath p r We now take the products A 2 p 2> A x q 1} the sum of 

 which gives us A 3 = 1. Thereafter we take the sum of A 3 ^? 3 , A 2 q 2 , A 1 r t , to 

 obtain A 4 = 2. In this example the first five q's happen to be zeroes, and so 

 the middle terms of the expressions are awanting ; the middle term is first 

 found in the expression for A 8 , which is A 7 .p 7 + A e . q 6 + A 5 . r 5 = 18 + 7 + 3; 



