316 
Proceedings of the Royal Society 
We have 
When <p is a constant scalar, i.e., when the resistance is in the 
direction of motion (which is the case generally in physical appli- 
cations) the middle term vanishes, and we have 
In fact, in this case, <p and % are commutative in multiplication, 
so that the equation in ^ may be solved as an ordinary quadratic. 
Even this very particular case involves a singular question, 
though not one of such difficulty as that of the general problem 
above. We have, in fact, to solve an equation of the form 
where w is a given, and nr a sought, linear and vector function. 
This leads to an equation of the sixth degree in with pairs of 
roots equal but of opposite signs. The coefficients of the cubic in 
33- are formed by the solution of a biquadratic equation.* 
* Suppose the cubic in -nr to be 
-nr 3 + g-zr 2 + ggur + g» — 0 , 
the given equation enables us to write it in either of the (really identical) forms 
(nr + g) u + gga + g 2 = 0 , 
or *r(« + gi) + ga + g 2 = 0 ; 
or, as it may be written, 
whence 
(g <* + g * 
V » +ff! 
or 
w 3 + (2 g x — g 2 ) «2 + {g\ - 2 ‘ggja - g\ — 0 . 
If the cubic in a be 
a 3 -f mu 2 + + m 2 — 0 , 
we have by comparison of co-efficients 
2 g l —g 2 — m, g\ — 2gg t = m x , g\ — ~ m 2 
so that g 2 is known and 
