336 



DR THOMAS MUIR ON THE 



(2) the biaxisymmetric determinant 



6 Jxy 

 a 



fj¥» 

 c Jxz 



d Jxw 

 h Jxyzw 



9 Jzw 



and (3) the axisymmetric determinant 



b 



a 



c Jxz 

 d Jxw 



ejyz 



fjyw 



g Jzw 

 It Jxyzw 



c Jxz 



cjyz 



a 



y Jzw 



b Jxy 



h Jxyzw 



1 1 Jxw 



fj~y™ 



d Jxw 



fjy™ 



(J Jzw 

 a 

 h Jxyzw 

 b Jxy 

 c Jxz 

 ejyz 



ejy* 



c Jxz 



b Jxy 



h Jxyzw 



a 



g Jw> 



fJn' r 



d J.no 



fjyw 



d Jxw 

 h Jxyzw 



bj^i 



y Jzw 



a 



ejye 

 c Jxz 



g Jzw 



h Jxyzw 



d J, I ir 



c Jxz 



f Jy w 

 ejyz 



a 

 b Jxy 



a 

 bxy 

 cxz 

 dxw 

 eyz 



fyw 



yzw 



a 



ez 



fw 



cz 



c 

 ey 



a 



gw 

 by 



d 



fy 



a 

 hyz 



dw hyw by 

 hzw dw cz 



e 



ex 

 bx 



hxw 

 a 



gw 



fw 



f g h 



dx hxy g 



hxz dx f 



bx ex e 



gz fy d 



a ey c 



ez a b 



h Jxyzw 



g Jzw 

 fjyw 



e Jyz 

 d Jxw 



c Jos 



b Jxy 



a 



hxyzw gzw fyw eyz dxw cxz bxy "'a 

 The third form is easily changed into the second by multiplying the columns in order by 



1> Jxy> J xz > sl xw > Jyz, . . . • , Jxyzw, 



and then dividing the rows in order by the same. The mode of resolution of the second 

 form into factors is well known.* 



16. There is still another variant of the problem of sections 6, 8, viz., to express j 



the relation 



cos- 1 ^ + cos -1 ?/ + cos _1 z + cos _1 w = 



in purely algebraical form. In essence it is the same as the variant dealt with in 

 section 13. 



Subtracting cos -1 ;/; from both sides, and then taking the cosines of the two equals 



we have 



xyz - xjl-y 2 Jl-z 2 - y Jl-z 2 Jl-x 2 - z Jl-x 2 Jl-y 2 = w, 



which is at once seen to be an equation of the form dealt with in section 12. The resul 



of the rationalization is 



xyz w + 2xyz 



y x w z + 2yxw 

 z w x y + '2zwx 

 io z y x+2vjzy 



= 0, 



or 



2» 4 — 2H,x 2 y' 2 + Sxyzw + 42a% 2 « 2 — 42 x?yziv = , 

 * See Quart. Journ. of Math , xviii. pp. 170, 171. 



