316 
Proceedings of the Ptoyal Society 
We have 
+§(V..p-pV.«)-V.cV.c. 
When 9 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 
or, as it may he written, 
In fact, in this case, <p and ^ are commutative in multiplication, 
so that the equation in ^ may he 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 to is a given, and zr 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 
35 - are formed by the solution of a biquadratic equation.* 
* Suppose the cubic in ct to be 
+ gTx 1 + + g* — 0 , 
the given equation enables us to write it in either of the (really identical) forms 
or 
+ g)u + g x -a + g» = 0 , 
w(« + ffj) + go + ^0 = 0; 
whence 
or 
If the cubic in a be 
<y 3 + mu- + m x u + m 2 = 0 , 
