diff --git a/inc/model_results/lens_charge/lens_charge.tex b/inc/model_results/lens_charge/lens_charge.tex
index bcd93a6..77b64c9 100644
--- a/inc/model_results/lens_charge/lens_charge.tex
+++ b/inc/model_results/lens_charge/lens_charge.tex
@@ -18,7 +18,7 @@ \section{Space-charge Effects and Spatial Focusing} \label{sec:mag_lens_charge}
 
 The single-electron ($ N = 1 $) focal spot sizes of $\sim$150 microns for the $ f = 60 \text{mm} $ lens and $\sim$15 microns for the shorter $ f = 6.0 \text{mm} $ lens show marked increase starting around $ N \approx 10^{ 5 } $ and $ N \approx 10^{ 6 }$, respectively, and worsen dramatically above these thresholds.
 This order of magnitude difference is fundamentally related to the greater time of flight before the focus of the $ f = 60\text{mm} $ lens; it increases the impulse that the internal Coulomb force can exert on the pulse as its charge density is increased under focusing.
-A consequent further effect is a shift in the focal position to $ z > f $.
+A consequent further effect is a shift in the focal position to $ z^{\prime} > f $.
 A more subtle effect is that oblate pulses are more readily focused than prolate pulses --- the rate of transverse pulse broadening due to space-charge effects already being intrinsically greater in the latter (\ref{fig:compare_shape}).
 
 %TODO? include simulations for different voltages to establish TOF argument?
diff --git a/inc/model_results/lens_charge/lens_charge_plot.tex b/inc/model_results/lens_charge/lens_charge_plot.tex
index 33ae177..9486aeb 100644
--- a/inc/model_results/lens_charge/lens_charge_plot.tex
+++ b/inc/model_results/lens_charge/lens_charge_plot.tex
@@ -45,7 +45,7 @@
     Each plot's pulses begin at the lens' front focal plane at velocity $c/3$ (related to 30 kV energy) and have the same volume.
     The left plots (\subref{fig:focus_long_oblate} and \subref{fig:focus_short_oblate}) are initially oblate shaped, HW1/eM width $ 500 \mu \text{m}$ and ellipticity $ \xi ( 0 ) = 0.1 $.
     The right plots (\subref{fig:focus_long_prolate} and \subref{fig:focus_short_prolate}) are initially prolate shaped, HW1/eM width $ 107.7 \mu \text{m}$ and ellipticity $ \xi ( 0 ) = 10 $.
-    Distance traveled in column is measured relative to the lens at $z=0$, in units of the focal length.
+    Distance traveled in column is measured relative to the lens at $z^{\prime}=0$, in units of the focal length.
     The top plots (\subref{fig:focus_long_oblate} and \subref{fig:focus_long_prolate}) have a longer focal length $f = $ 60mm compared to the bottom plots (\subref{fig:focus_short_oblate} and \subref{fig:focus_short_prolate}) $ f = $ 6.0mm.
     A logarithmic scale is used for comparative clarity near the foci.
     Clearly the best performance is achieved for shorter focal lengths and oblate pulses \subref{fig:focus_short_oblate} even at higher initial charge densities.
diff --git a/inc/model_results/spacecharge/spacecharge.tex b/inc/model_results/spacecharge/spacecharge.tex
index 33ff751..496db37 100644
--- a/inc/model_results/spacecharge/spacecharge.tex
+++ b/inc/model_results/spacecharge/spacecharge.tex
@@ -50,7 +50,7 @@ \section{Effect of Space Charge on Dynamics} \label{sec:free_spacecharge}
 It is immediately obvious that the final transverse width is larger than in the ideal case presented in \ref{fig:spacecharge_noacc}, even in the low charge-density regime; this is primarily due to the additional divergence imparted by the negative lensing at the anode (see Section \ref{sec:gun_model}).
 One can see that though the pulse size does increase beyond a certain total pulse charge, and while that trend appears to start at a higher charge, due to the complexity of the pulse dynamics it is rather difficult to compare the exact charge-density from this plot to that in the idealistic case.
 What can be seen is that the transverse size increases more sharply with pulse charge at higher levels.
-Since the pulse is already diverging in the transverse direction, this is compounded by the increased pulse charge.
-This divergence then acts to lower the overall charge-density more quickly, thus the longitudinal pulse length is not affected as strongly as was seen in \ref{fig:spacecharge_noacc}.
+Recall that the pulse is already diverging in the transverse direction, the increased pulse charge now serves to compound this action.
+This increased divergence then acts to lower the overall charge-density more quickly, thus the longitudinal pulse length is not affected as strongly as was seen in \ref{fig:spacecharge_noacc}.