Transformation of variables in definite integral limits

Hello,

I am going through Whittaker's treatise on Classical Mechanics. In chapter 3 he derives the equation of motion for a simple pendulum, and I have a question about his method.

Starting from the general form for the equation of energy (s is the path):

[itex]\frac{m}{2}\dot{s}^2 = \int\limits_{s_0}^{s} f(s)\, ds + C[/itex]

and using

[itex]s = a\cdot θ[/itex]

and

[itex]f(s)=-m\cdot g\cdot sin(θ)[/itex]


The equation of energy is transformed to:

[itex]\frac{m}{2}{(a\cdot \dot{θ})}^2 = \int\limits_{θ_0}^{θ} -m\cdot g\cdot sin(θ)\,\cdot a\cdot dθ + C[/itex]

But if I remember my Calculus correctly, the transformation should be:


[itex]\frac{m}{2}{(a\cdot \dot{θ})}^2 = \int\limits_{a\cdotθ_0}^{a\cdotθ} -m \cdot g\cdot sin(θ)\,\cdot a\cdot dθ + C[/itex]

Notice the limits of integration. Where am I going wrong here? This is on page 72 here: http://archive.org/stream/treatisana...earch/pendulum

Thanks for your help.