Doubt in Maxwell's treatise I am reading Maxwell's treatise Vol 2 page 154 and it seems to me that something is incorrect. 

Equation (9) is fine. Now let's derive equation 10 from this.



A 1954 edition of this treatise (called Dover edition) has $+2C$ inserted but in the forthcoming equations, $+2C$ has been taken away for no reason.

So my question is: Should we really replace the highlighted $-\alpha' (\hat{r})$ with $\alpha' (\hat{r})$? If so, then for what reason?

 A: In short, your derivation is correct, and you are also correct in saying that the Dover edition incorrectly drops the additional $+2C$ later on.  However, the end result is not affected.
The edition you show a picture of at the beginning of the question is the first edition (1873), and the Dover version is the third edition (1891), so that's how I'll refer to them.  You can see that the relevant text was not changed in the second edition.  J.J. Thomson revised the third edition.  He said in the preface that he had made some changes including "[verifying] some results which Maxwell gives without proof; I have not in all instances succeeded in arriving at the result given by him, and in such cases I have indicated the difference in a foot-note."  Evidently, he took additional liberties when he convinced himself that there was an outright error — by simply changing the text without any note.
The relevant change can be made to the first edition starting from equation (10), by simply replacing $A$ with $A+2C$.  Unfortunately, the third edition dropped the parentheses that would be needed to do this correctly in equations (10) and (11), but that mistake doesn't carry through into equations (12) through (16).  Then equation (17) and (25) are missing the $+2C$ term.  However, it turns out that those two mistakes cancel each other out, so that equation (26) and the following are not and should not be changed.
So it's fair to say that Thomson (or at least his publisher) made some mistakes.  But it also seems that Maxwell made some mistakes.  On the previous page, Maxwell says

We shall also consider $ds'$ resolved into $\cos \theta'\, ds' = \alpha'$ in the direction of $r$ reversed...

It's strange wording to modern ears, but I think you're entirely correct to interpret that as meaning $d\vec{s}' = - \alpha' \hat{r} + \ldots$, as opposed to $d\vec{s}' =\alpha' \hat{r} + \ldots$.  But it's also true that the $+2C$ disappears if you go with the latter.  And that's what you would get if you measured $\theta'$ in the same way as $\theta$ — meaning you would take the complementary angle instead, and nothing else in the problem would change.  Both of these conventions are entirely reasonable ways to set up the problem.
My best guess is that one of two things happened:


*

*Maxwell simply made a sign error somewhere in his math, or

*Maxwell chose one convention originally, changed mid-stream, and forgot to go back and correct everything.


Either one of these is quite possible.  For example, Maxwell didn't deal very clearly with the signs of forces.  He talked about "mutual action", the "action between the components", and "the force between them".  In modern times, we keep the signs straight by talking about the force of one on another — and we specify which is "one" and which is "another".  Maxwell's approach is prone to changing your mind.  Obviously, it is pure conjecture to extend that to claiming that this is the source of Maxwell's error, but that is all we have unless we can find contemporaneous communications or notes (e.g., Thomson's).
In any case, it's reassuring to know that two of the greats of physics made the same kinds of mistakes I do.
