273 
1909-10.] Dr Muir on the Theory of Orthogonants. 
priori for the appearance of these coefficients here.” We are thus not 
wholly unprepared for a communication from Cayley himself on the 
subject of the construction of a linear substitution for the transformation 
of oc{-\-x\-\- . . . into £? + $ + .... The following is his procedure, four 
variables being used in place of his n. 
With unity and any six quantities whatever there is first formed the 
square array 
^13 h 
^23 ^2 
Zn ^10 ^10 Z- 
12 
13 
or say 
21 
-l. 
v 23 
'24 
34 
^34 
1, 
Z 22 ^23 
l 
l 
^24 
Zoi Z q0 Zq 
31 
to Z 
42 
43 
44 
where l rr = 1 and l rs = —l sr . Then taking a new set of four variables 
0i , # 2 , d 3 , (9 4 , and using for their coefficients the quantities in the square 
array, firstly as disposed in rows, and secondly as disposed in columns, he 
puts 
and 
^11^1 ^12^2 "** ^13^3 ^14^4 
= 
x 1 
^21^1 "t ^22^2 ^23^3 ^24^4 
S 
x 2 
^31^1 + ^32^2 "t ^33^3 "t ^34^4 
= 
x 2 
^41^1 + ^42^2 "h ^43^3 + ^44^4 
= 
X 4 , 
Zn^i + Z 21 0 2 + Z 31 0 3 + Z 41 # 4 
= 
fiP 
^12^1 ^22^2 "h ^32^3 ^42^4 
= 
^2 
^13^1 "h ^23^3 "h ^33^3 + ^43^4 
= 
t. 
^14^1 "t ^24^4 ^34^3 ^44^4 
= 
fj 
thereby ensuring that + x 2 2 + . . . = £ 2 + £ 2 2 -f . . . Solving the two sets 
of equations separately for each of the 0’s and equating the results he next 
obtains 
^11^1 + k 21 ^2 + k 31 £C 3 + L 41 iT 4 = Ljjfj + L 12 ^ 2 + ^ 13^3 + L 14 £ 4 
Li 2 ^i + L 22 a? 2 4 * L 32 o? 3 + L 42 * 4 = L 24 ^ + L 22 £ 2 4- L 23 4 - 1 - L 24 £ 4 
Li3^i + L 23 # 2 + L 33 ^ 3 + L 43 iT 4 = 4- L 32 | + L 33 £ 3 + L 34 | 4 
Lii^i 4- L 24 a? 2 + L 34 a? 3 + L 44 a? 4 = L 41 ^ 4 4 - L 42 f 2 + L 43 £ 3 4- L 44 £ 4 ^ 
where L rs is used for the cofactor of l rs in the determinant (A say) of the 
initial array. It only then remains to obtain from this the os’s in terms of 
the £’s, or the £’s in terms of the x’s. This Cayley does by using as multi- 
pliers, in the former case the elements of any row of the original array, and 
in the latter case the elements of any column. Thus, multiplying by l n , l 12 , 
l ls » l \4 respectively and adding he obtains 
Ax 1 = (2Z n L n - A)^ + 2Z n L 12 £ 2 + 2Z n L 13 £ 3 + 2 Z 
11 ±J 14^>4 5 
