634 ^^ WALLACE ON A FUNCTIONAL EQUATION. 



now AEB has been proved equal to ADC, and by construction KAE = BAD ; 

 therefore the figures BADC, KAEB are equiangular ; they are also made up of 

 similar triangles, therefore they are similar. 



In the same way it may be proved that the figure BADC is similar to E'AHD, 

 the equal angles being E'AH = BAD and AHD = ADC, HDE' = DCB and AE'D = 

 ABC. 



Thus it appears that the three figures BADC, KAEB, E'AHD are similar, and 

 that the lines AB, AC, AD have to each other the same ratios as the lines AK, 

 AB, AE have to each other ; also the same ratios as the lines AE', AD, AH, have 

 to each other. And since, by hypothesis, a force represented by AC, and acting 

 on the point A in the direction of that line, is equivalent to two forces represented 

 by the lines AB and AD, "acting at A in their directions ; so, by reason of the simi- 

 larity of figures, a force represented by AB, and acting in its direction, will be 

 equivalent to forces represented by AK, AE acting in the direction of these lines. 

 Also a force represented by AD, and acting in the du-ection AD, will be equivalent 

 to forces represented by AH, AE' acting in the directions of AH, AE'. 



It now appears that the force expressed by AC, which is the resultant of 

 the two forces AB, AD, may also be considered as the resultant of the forces AK, 

 AH, together with the resultant of the equal forces AE, AE'. But the force AK 



A B^ P^ A D^ 0,2 



is by construction ='T~n = ^ ; ^^^ the force AH= .^ ~~r"' ^'iid the two equal 



AB AD 

 forces AE, AE', which are each equal to t^ — , and make with AC angles each 



equal BAD=a, have been proved by our first case to compose a force equal to 



2PQ, 

 2AE . cosEAC= — ^^ — cos a; therefore, on the whole, 



^ P2 Q2 2PQ 



and R2=p2 + Q2 + 2PQcosa. ..... (B) 



Now, by the elements of geometry, this last expression is the diagonal of a paral- 

 lelogram whose sides about one of its angles are P and Q, and the contained 

 angle « : hence we have this proposition. 



Theorem. — Two forces which act on a point hi the directions of the sides of a 

 parallelogram, and which are represented in magnitude hy these sides, are equiva- 

 lent to a single force acting in the direction of the diagonal, and repyresented in magr 

 nitude hy that diagonal. 



In this way, by the theories of analysis and Geometry, the proposition which 

 is the foundation of statics is derived from a few axioms, wliich are analogous to 

 those of Geometry, and which seem to be necessar}?^ consequences of our primary 

 notions of a force. 



