Application of Undulatory Theory. B/j A. E. Conrady. 407 



In this parallelogram we have the angle F C D = (/3, — /3 2 ), 

 hence angle D E = 180 - (/3, - /3 2 ). 



In the triangle C I) E there are therefore known side C D = F 2 , 

 side DE= F Xj and the included angle ODE = 180 - (& - /3 2 ). 

 To tind the third side we apply one of the fundamental equations 

 of plane trigonometry (a 2 = b- -f c 2 — 2 o c cos a), which gives — 



F = x/Fi 2 4- F 2 2 - 2 F x F 2 cos {180 - (fr - &)} 



but for any angle we have cos (180 — a) = — cos a, hence — 



F = */¥{> -4- F 2 2 + 2 F x F 2 cos (ft - &)" 



which becomes identical with (6) if letter F is changed to A. 



If we assume the angle j3 between 

 C Z and the resultant to have also been 

 determined, the triangle C D E gives us 

 another interesting relation, for as angle 

 E C D = (/3 - £ 2 ) and angle DEC 

 = F C E = (/3x — J3), we have, remem- 

 bering that cos (a — /3) = cos (/3 — a) — 



F = Y 1 cos (£ - A) + F 2 cos (/3 - /3 3 ) 



an interesting relation which may be 

 used — when applied to amplitudes — to 

 check the accuracy of a calculation, but 

 which is in no sense a solution of the 

 problem, as it requires the phase- 

 relation between the resultant and the 

 components to be known. 



It is very important to remember when making use of this 

 relationship between the combination of forces and that of 

 amplitudes, that the angles yS have really a totally different 

 significance in the two cases ; in the case of forces they really 

 measure angles between the direction of forces, whilst in the case 

 of amplitudes the direction of the disturbances is always the same 

 (up and down in our figures), the /3 measuring the difference of 

 phase in the sense of our fig. 75. 



The formulae (6) and (7) can easily be extended to combine any 

 number of disturbances. The formula corresponding to (6) 

 becomes the square root of a squared polynomial in which each 

 double product has a corresponding cos (^„ — /3 m ) as factor, and 

 the equation for tg /3 becomes the quotient of the sum of all 

 terms A m sin fi m divided by the sum of all terms A wl cos /3 m . In 

 concrete cases, the numerical determination is really simpler if the 

 equations (4) with the proper number of terms have their right 



Fig. 76. 



