REVIEWS — A TREATISE ON ANALYTIC AX STATICS. 301 



mark will apply to all the succeeding steps, unless the additional 

 condition which we have indicated he introduced. This condition 

 really is introduced in the question before us, by mechanical consi- 

 derations — by the assumption, in fact, that the direction of the resul- 

 tant of two forces necessarily lies in the angle contained by the di- 

 rections of the forces themselves. From this it will follow that so 

 long as x is not greater than 90°, f(x) is positive: so that a standing 



for 60°, it will follow that /( — ) and cos ( — ) being both ne- 

 cessarily positive will have the same sign. Thus for example we should 

 get from the data 



If (30°) i 2 = i cos '30 



or/ (30°) = + 2 cos 30 



and the mechanical considerations justify us in taking the upper 



sign. And it is easily seen that though it is true that f (60°) = 2 



cos(5 x 60°), yet when the additional mechanical considerations are 



taken in. the above proof will not serve to shew generally that/ (x) 



= 2 cos 5 x. In fact, if these considerations are fairly introduced, 



the proof becomes perfectly unexceptionable. 



G. C. I. 



NOTE ON 



PoissoiCs Proof of the Parallelogram of Uorces* 

 The general functional equation, from which Poisson obtains his 

 solution by an indirect process, may be treated directly as follows : 

 The equation is 



If we expand in ascending powers of z, by Maclaurin and Taylor's 

 theorems, we obtain 



/O) {/(0) +f(0)z +f"(o) z l + I 



= 2/(*) + 2/"(*) + 



Equating corresponding coefficients of z, we have 



/O)/(0) = 2/(*) ; 



which is satisfied either by/(#) = 0, or/(0) = 2. 



Confining our attention at present to the latter solution only, and 

 proceeding to equate coefficients, we find/' (0) *= 0, and all the suc- 

 ceeding derivatives of an odd order also vanish. Also we have 

 2/'»=/"(0)/(*); 



* Vide No. 1, Reviews, " A Treatise on Analytical Statics," &c., ante, p. 63. 



