1896-97.] «*• Prof. Tait on Equations in Vector Functions. 503 
where ra‘ is the product of the scalar roots of 57-, and therefore 
m — m- Se57c = m + e 2 . 
[The former solution, giving 
X p = o7- ; to57~- ; p 
= p cos A + sin A 57 aTX~ l p - ( 1 - COS A)67-aSa57 ~-p , 
contains this as a particular case, for it is easy to see that the two 
expressions agree if we are entitled to assume simultaneously 
cos A = 1 
2e 2 . A 2 era 
— , sm A = , 
m m 
1 - cos A = 
2e 2 
m 
The first and last are identical ; and the first and second require 
merely that we shall have 
1 = 
2e 2 V 2 4e 2 m 
) + 9 ; 
m ) 
which is satisfied in consequence of the expression for m above.] 
That the complete admissible value of x is what we have already 
found, and contains only the one scalar indeterminate A, is easily 
verified by expressing ^ as a linear combination of the operators 1 , 
57"Va57~" , 57-aSa57 - -, which are suggested by its relation to 
and are obviously commutative with one another; and indepen- 
dent, in the sense of not producing any new operator by their 
combinations. Then the required relations among the coefficients 
are determined by comparing term by term the expressions for 
<t>X and • 
9. Finally, we may treat (5) by a method similar to that adopted 
for (1). Let a now be a directional root of y, so that ^'a=q/a. 
Then we have 
X cf)a = 
— <f) a . 
9 
But the cubics of ^ and are necessarily identical, and thus 
their common numerical roots can be no others than 1 , g, 1/p. 
Also, since (/> is assumed to be real, g is imaginary, for </> changes 
the g directional root of x to the 1/p root of y, and conversely. 
But, if we operate by the conjugate of (5) upon a, we get 
X c£'a 
1 
— <p a . 
9 
