358 Sir James Cockle on Transformation: 



transformed equation is free from x n . Put a,e = l and h, k=0; 

 the factors are in arithmetical progression, as, by art. 13, they 

 should be. Put A,E = 1, 2; then art. 19 gives 



P 2 =0. Q 2 =4 + 9M-12L, R 2 = 32 + 18N-3M, 



' S 2 = 6N + 12. 



Hence 

 and 



2B 2 =16 + 12(L-M + N) = 16 + 12Z>, 



6C 2 =12L-6M or C 2 =6 + L-N; 

 and the first two conditions of art. 20 are satisfied, as is the 

 last; for, by art. 18, b 2 = b + 2. 



56. Since n and ?? 2 are to be taken as one-valued when the 

 quantities in terms of which they are expressed are so taken, 

 we infer that the radical in arts. 10 and 17 means the appro- 

 priate root, not either root indifferently, and that the root can 

 be rationally expressed. This is the case ; irrationality has 

 disappeared from all our results. 



57. Multiply the products «xa 2 and &&, wherein the factors 

 are in order of magnitude, into 7. If we preserve the order, 

 we get six configurations, viz. ya^, uf/ct 2 , u x a 2 y, and yfii@ 2 , 

 fifypt, /3i/3 2 7« Assume that there is a value of y which will 

 make one of the a set and also one of the /3 set an arith- 

 metical progression. We get nine mutually exclusive but 

 severally possible pairs of relations, of which one, for example, 

 is 



« 2 + 7 =2 ai , 02 + 7=2/3! ; 

 whence 



4« 1 -2« 2 =4/3 1 -2/3 2 or3I=3J + U. . . (14) 

 Let a be taken to represent D — a, and we thus obtain by 

 Boole's algorithm a result included in those of art. 50. The 

 same thing holds for the remaining eight cases. 



58. This last verification indicates that a factor always dis- 

 appears from (13), and that the solution t] 2 = — 1 must be ex- 

 punged from art. 53; for, by art. 36, co = 0. In the numerator 

 of (13) of art. 50 change D into D — 3, or, rather, omit the 

 transformation by which D has replaced the D — 3 which would 

 otherwise have remained there. A factor will disappear if 

 any one of the nine expressions 4<u + 3(J + I), 4« + 3I, 

 4&> + 3J, co vanishes. But these conditions coincide with those 

 of arts. 33, 34, and with those implied in art. 57. 



59. The introduction of the arithmetical progression, while 

 it affords a good verification, restricts generality. Putting 



B=A-i, A = D + « 2 -l, z= x a >X~\ 

 /(A,B,I) = (A + 2B)(A-B + fI)(A-B-fI), 



