I have seen formulas out their for adding 2 lenses with oblique crossed cyls, but I had never seen one to subtract them and the formula in Stoners book doesn't play nicely backwards so after a little looking around I found this:

To Subtract a Rx from another

Rx1-Rx2=Rxtotal

Rx1 would be comprised of S1 C1 a1

Rx2 would be comprised of S2 C2 a2

So to subtract on Rx from the other we would need to break down the Rx into 3 components the Mean Refractive Error (MRE) just like it sounds the average between both meridians of power:

MRE=S+C/2

The cylinder power on the 180o meridian

C(0)=C*cos(2a)

and the cylinder on the 45o meridian

C(45)=C*sin(2a)

These two components can be added or subtracted to find the sum of two lenses, so in our case we are going to subtract:

MREtotal=MRE1-MRE2
MREtotal=(S1+C1/2)-(S2+C2/2)

C(0)total=C(0)1-C(0)2
C(0)total=C1*cos(2a1)-C2*cos(2a2)

C(45)total=C(45)1-C(45)2
C(45)total=C1*sin(2a1)-C2*sin(2a2)

Ok so now that we have subtracted our 3 components to convert it back to shpero cylindrical form we would:

Ctotal=Sqrt[C(0)total+C(45)total]

tan(2atotal)=C(45)total/C(0)total

Stotal=MREtotal-Ctotal/2

Alright and thats it. Again just another exercise for the mind.