Invariance of Mass in SR: Comparing Taylor and Wheeler's Spacetime Physics with a Comment made in Purcell and Morin's Electricity and Magnetism Purcell and Morin's Electricity and Magnetism mentions on pp. 241:

Mass is not invariant in the same way (in a relativistic theory). We know that the energy of a paritcle is changed by its motion, by a factor of
$$ \frac{1}{\sqrt{ \displaystyle 1 - \frac{v^2}{c^2}}}.  $$

This is all right; I can see the relationship this is alluding to. But as it is mentioned in Spacetime Physics by Taylor and Wheeler,

... $m$ is a consntant -- the same at all speeds, all places and all times. Any difference between the spacetime formua for momentum and the corresponding Newtonian formula is therefore attributed to the difference between proper time and labrotary time.

I'm slightly confused. Are both statements compatible with each other? Taylor and Wheeler derive the following formula for relativistic momentum,

$$ p_{R} = m \frac{dx}{d\tau} = m \sinh\phi, $$

such that

$$ p_{R} = p_{N} \cosh\phi, $$

where $p_{N}$ denotes the Newtonian formula.
 A: 
Purcell and Morin's Electricity and Magnetism mentions on pp. 241: Mass is *not* invariant in the same way (in a relativistic theory). We know that the energy of a particle is changed by its motion, by a factor of 1/${\sqrt{1 - {v^2}/{c^2}}}$

Purcell and Morin are correct here. Their figure 5.7 features two massive particles rotating on a pivoted rod in a box. Because they're moving, the mass of the system is increased. 

This is all right; I can see the relationship this is alluding to. But as it is mentioned in Spacetime Physics by Taylor and Wheeler ... $m$ is a constant - the same at all speeds, all places and all times. Any difference between the spacetime formula for momentum and the corresponding Newtonian formula is therefore attributed to the difference between proper time and laboratory time.

I haven't got the book and I can't find it online. I imagine that as knzhou said, this is referring to rest mass in a special relativity context, wherein the rest mass of your charged particle doesn't change. However as Einstein said, special relativity "is nowhere precisely realized in the real world". In the real world the mass of the electron changes when you drop it. Gravity converts potential energy into kinetic energy. When you stop the electron, this is radiated away. The electron is then left with a mass deficit.  

I'm slightly confused. Are both statements compatible with each other? 

Sorry, I'm not sure. The rest mass of each charged particle is unchanged when you spin them on the pivoted rod. But adding energy of the contents of the box increases the mass of the box. It's similar for the photon in the gedanken mirror-box. Catch a massless photon in a mirror-box, and you increase the mass of that system. See http://arxiv.org/abs/1508.06478 by van der Mark and (not the Nobel) 't Hooft. When you open the box, it's a radiating body that loses mass, just like Einstein's E=mc² paper. Also see this article by physicsFAQ editor Don Koks. Part of the problem is that mass is rather ambiguous. Nowadays when we say mass without qualification we mean rest mass, but that wasn't always the case, and it's important to distinguish between rest mass, relativistic mass, inertial mass, and so on.  
