Comparing two time domain signals in a scale and shift-invariant way I have two signals in the time domain, call them $S_1(t)$ and $S_2$(t). Because of calibration issues relating to the underlying devices that these signals are obtained from, they will not necessarily have the same scale or mean. What is the appropriate way to compute a measure of similarity such that, if $S_2(t)$=$C_1 * S_1(t) + C_2$, where $C_1$ and $C_2$ are constants, my measure of similarity will say that these signals are identical? Or at least indicates that they are very similar?

Edit: The solution provided by Digiproc is indeed a good solution to the problem as I originally stated it. However, it seems that I simplified my original problem too much for this question and that their solution would not work in my case. In particular, one of my signals has noisy spiky behavior in addition (i.e., very sharp and large localized peaks due to sensor technical issues). Therefore the dividing of one of the samples by the sample magnitude is not going to lead to a good alignment of the two signals. Is the MSE difference approach mentioned by 
robert-bristow-johnson in the comment the best approach then?
First remove the DC offset (C2) from each signal by subtracting the average level from each sample, then removing the arbitrary scaling (C1) of each signal by dividing each sample by the average sample magnitude.