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 4, Note on Professor Tait’s ‘“‘ Quaternion Path” to Deter- 
 minants of the Third Order. By the Rev. Hugh Martin, 
 M.A. Communicated by Professor Kelland. 
 
 I have read with much interest Professor Tait’s ‘“‘ Note on Deter- 
 minants of the Third Order” in the Proceedings of this Session 
 (pp. 59-61), and admire the method of discovering new proper- 
 ties of Determinants. I am not sure, however, that the properties, 
 when discovered, are more difficult of proof by Determinant 
 methods, and I venture to submit the following as simple and 
 elementary :— 
 
 The first property, namely, 
 
 ete, y ty 2+%|=2 
 
 He YAY 4% 
 
 te y+y +2 
 is true under greater generality, and the Determinant proof is the 
 same as for the special case. 
 
 VOL. VI. »41950 
 
Proceedings of the Royal Society 
 
 ° 
 
 ‘ S1OYJO OY} JO YOVO WOT MOL SIF MOU SUI}[USEI OTT} JORAIQNS : yUvUTTIOJOP OY} OpIsyno (T—u) OJON oy} Oye 
 
 , uly 
 
 po: stp (qu) ie: OTD see +1 +(q-u) 
 
 ‘ATOUVU ‘MOL ISIE MOU B WIOJ OF SMOI OY} [IP PPV 
 
 xt aos —u* —u' 5) ut 
 ce Op ee ot Pees ee eee ik 0 ware ae: yee Se 10) 
 
 ° ° e e e e e e e ° e e ° ° ¢ e ° 4 e e e e ° ° e ° ° ° ° e 
 
 ° 
 e 
 . 
 eo 
 e 
 
 e ° e e ° e e e e e e e (4 . . ¢ ° e ° ° e ° ° 
 
of Edinburgh, Session 1866-67. 123 
 
 then add to it the sum of the others: and we have 
 
 =(n-1 “ves 
 ¥ ( ) Oe Cn Cn 
 a eens Br to 1 cy Ont 
 ie oa g 7 ay 9 Ki a, 2 
 or 6 ay is a Poe ah std Sasa: oe 
 
 The negative signs may be removed and compensated for by the 
 factor (—1)"—' placed outside the Determinant, and the first row 
 may be made the last by multiplying again by (—1)"—', which 
 
 counteracts the former multiplication. Hence 
 
 V=(n-1) @.7 % 1° oar a, 
 
 When n=3, this is the quaternion theorem in question. 
 
 The second formula is the well-known property of the reciprocal, 
 and a simple proof of it will be given below. 
 
 The third and fourth are immediate translations of Quaternion 
 expressions into Determinant forms. And I suppose there is no 
 analytical difficulty in enlarging the number of Sir William R. 
 Hamilton’s symbols, 2, 7, & (though it would introduce the concep- 
 tion of ideal space of more dimensions than three), and thus ex- 
 tending the reach of the “ Quaternion path” to Determinants of 
 
 any order. 
 
 The jith formula, in its algebraical expression, would be 
 
 y2z2i|e | y, 2, ze |+| 2%, & = —-2|u yz |? 
 Y By Y¥2 oP) aN w oP) , ‘ FS w, 4, a 
 
 a Ya Be 
 9% | F | YY 4, & & 7 
 Yo %, ET x, ms 
 Y.% | bly % 
 ye ye, |? Ce &e. 
 
124 Proceedings of the Royal Society 
 Now, using capitals for minors we have, by the first property, 
 
 Sak Y.42y. vee 2[ Xi Bea 
 
 Ko RY YZ ee xX, ae 
 X, +X Yt Y, Z+4Z, 2 cea 
 a PR a —2\o 9 2 
 = XA = © i ae Q. E. D. 
 Ay Xe Ay %, Y, %, 
 
 This, of course, may be generalised like the first formula; and 
 the result is 
 
 se + ee De rae . +A, 9: &., &e. 
 
 ee ao Ae A. at : +A Ue &e., &. 
 ae bt AD h A, ih ° FA, » &.; &C. n / 
 
 Ai ig ye Ay or A, i+ ; = A &c., &e. 
 (es —1 
 (n — 1) Oy ig Sir 4) n 
 oe . e e ea 
 
 The sixth formulais got by a reduplication of the proof of the 
 Jifth,whether the method used be Quaternion or Determinant. 
 
 As to the reciprocal, the following seems quite elementary :— 
 Let 
 
 Y%, — 24, 2h, — LZ, LY, — YX, 
 Yi%, — 2 Yn 20, — U2, VY, — YX |= 4 
 Ye — ay Be — m2, BY — Y~ 
 
 Multiply the first row by «,, the second by a, the third by w,; add 
 all the rows for a new first row; divide the second row by @, and 
 the third by «,. 
 
 q 9% 
 £, Y, 2, 0; 33920 
 £9 2 = %,V 
 xX Vigo 4 
 X, Vee: 
 
of Hdinburgh, Session 1866-67. 
 
 Hence 
 Pat YS 
 wv, Yi zy y z,\=%¥ 
 Be Oates itis 
 
 But it is at once seen that 
 
 le 
 
 a, 4, # 
 1 1 1 
 af ZL, 
 ® Y, % 
 Hence 
 ae y ¢ |? 
 ae Bee Fee. 5 Q. KE, D 
 $. Y, %, 
 
So bo a sore ey bee 
 
f 
 
 Makers 
 Syracuse, N.Y. | 
 PAT. JAN. 21, 1908 
 
 Gaylord Bros. 
 
 Oye: <8 . . sped 
 Ral ke » ‘ 33 
 
 mae oa ae em ah et a ee 
 (est de awew ans teens ene we awte Sere ee see nwa we 
 Pose awsnse se. 
 
512.53T13NYM C001 
 NOTE ON PROFESSOR TAIT’S QUATERNION PATH 
 
 LE 
 
 3 0112 0170573