300 REVIEWS — A TREATISE ON ANALYTICAL STATICS. 



that if 2 x be the angle between two equal forces. P, and R their re- 

 sultant, we may write 



22 = P/(* ): 

 and that if 2 z be the angle between another pair of equal forces, we 

 shall have 



/(*) /(*) = /(*+*) + /(*-«) (1) 



and it is from this functional equation that the solution of the pro- 

 blem is to be derived. He notices that the assumption f(x) = 2 cos ex 

 satisfies this equation, and asserts that it is the only solution : an 

 assertion which is true only if it be understood that c may be either 

 a possible or an impossible quantity, and which, even with this modi- 

 fication his reasoning does not establish. What he does attempt to 

 shew is this, tbat if the particular assumption f ( x ) =2 cos x is 

 verified in two cases it must be true generally. That it is true when 

 x = is appareut from the equation itself by putting z — 0: an 

 appeal to mechanical considerations shews that it is also true when 

 x = 60°. The proof, then, to which we objected starts from these 

 data: that equation (1) holds: that/ (0) = 2 cos (0), and /(60°) 

 = 2 cos 60° : and from these data he professes to shew that / (x) 

 must be equal to 2 cos x for every value of x. "We objected to this, 

 that the very same reasoning might be employed to shew that f (x) 

 must be equal to 2 cos 5 x: and we inferred that the reasoning must 

 therefore be defective, and that the defect could be remedied only by 

 a fresh appeal to mechanical considerations. In effect it is not diffi- 

 cult to point out where this appeal becomes necessary. He first 

 shews that if the relation f (x) = 2 cos a? is verified when w=a it 

 will also be true when x is any multiple of a : he, then, has to shew 

 that it will also be true when x is equal to a divided by any power of 

 2. This is not generally true : it is true in the particular problem 

 we are solving : but as far as the data go this is not the case. In 

 order to make the proof hold generally, it would be necessary to add 



the word.-', " provided that we know from independent sources that 



>« 

 f ( — ) is of the same sign as cos — . Thus starting with the 



'— L 



known fact that f(a) = 2 cos a, he arrives at the equation 



\f{-) \'= 2 cos a + 2 



whence he at once infers that 



J (— \ = 2 cos — 

 J K 2> 2 



taking the upper sign in the ambiguity on extracting the root: in 



doing winch generally he is obviously not justified ; and the same re- 



