CHAP. XIX] RATIONAL ORTHOGONAL SUBSTITUTIONS. 531 



either diagonal shall be a, viz., 



(ac+bd}(pr+qs) =0, (ab+cd)(pq+rs) + (ad+bc)(ps+qr) = 0. 



He gave two special cases, one of which is his 161 above solution. 



G. R. Perkins 166 employed as the numbers of the first row of his square 



pp'+qq'+rr'+ss', pr'+qs'rp'sq', ps'qr'+rq'sp', pq'qp'rs'+sr', 



-pq'+qp'rs'+sr', -ps'+qr'-t-rq'sp', pr'+qs'+rp'+sq', pp' + qq'rr'ss', 



-pr'+qs'+rp'sq', pp'qq'-\-rr'ss', -pq'qp'-^rs'+sr', ps'-\-qr'+rq'+sp', 



ps'+qr'rq'sp', pq'qp'rs'sr', pp'+qq'+rr'ss', pr'qs'+rp'sq' 



those whose sum of squares equals (p 2 +? 2 +r* 2 +s 2 )(p' 2 -| ---- ). By writing 

 in reverse order the functions of the first row and changing the signs of 

 r, s in the first two terms and the signs of p, q in the last two terms, we get 

 the entries in the second row. We derive the third row from the first, 

 and fourth from the second, by moving each term one place to the right or 

 left without crossing the middle vertical column, and changing the signs of 

 q, s or those of p, r according as the term is moved to the right or left. 

 Two of the various possible such squares are given. Of the conditions re- 

 quired by Euler, 164 all are now satisfied except those relating to the two 

 diagonals. Take s = 0. The latter conditions become 



p'r' = q's' , p(p'q r r's'} r(p's' q f r'}. 

 By further specializations, he obtained the solution 



42+2? -11+4? 24- q 2-8? 



-18+8? -16+ q 24+4? 38+2? 



11+4? 42-2? 2-8? 24+ q 



16+ q -18-8? -38+2? 21-4?. 



C. Avery 167 proceeded as had Perkins, without describing the process to 

 choose the signs, and obtained the solution 



48+4? -44+3? 51-2? -7-6? 



-47+6? 21+2? 64+3? 12+4? 



44+3? 48-4? 7-6? 51+2? 



-21+2? -47+6? -12+4? 64-3?. 



The case ? = 5 yields Euler's 184 answer. 



V. A. Lebesgue 168 gave orthogonal substitutions in 3 variables in trigo- 

 nometric form. He 169 quoted Euler's 164 - 5 solution of the problem of 16 

 integers. 



L. Bastien 170 took four integers a, (3, y,8 such that ap/(y5) is the square 

 of rjs, where r, s are relatively prime integers. Write 



166 Math. Miscellany, New York, 2, 1839, 102-5. 



167 Ibid., 101. 



168 Nouv. Ann. Math., 9, 1850, 46-51. 

 i fi9 Ibid., 15, 1856, 403-7. 



Sphinx-Oedipe, 7, 1912, 12. 



