RESULTANT OF ELASTIC RESISTANCES. 



219 



to a or c. In either of those directions the displacement develops no deflecting 

 force; since, in the former, cosA=i and sin A disappears; and in the latter 

 shiA = l and cosA disappears. 



The arrows illustrate the relations and mutual action of these forces, and the 

 corresponding movements of the molecules. During compression the disturbing 

 force is CF, and the movement from C toward 1\ The opposing component 

 of the resistance is C'F, and the deflecting component GF. While CF pre- 

 dominates over C'F, the point of the arrow CF — that is, the direction of the 

 molecular movement — will be turned nearer the direction CD. But when C'F 

 predominates over CI', as in the return vibration, C'F represents the move- 

 ment, and the deflecting force turns the point of the arrow C'F nearer the 

 direction AC. 



The value of the resultant may be determined by means of those just given 

 for its components, from the right angled triangle CGH. For this gives us, 

 (putting p for the resultant, ) 



/)2=/-V2«V.os2A-F/2yVsin2A ; 



or p= A^fr Va^-os^A+c-^sin^A- 



The equation of the surface of elasticity also gives us, for the value of the 

 radial resistance (denoted by //) 



//=R2y;-=//-a3co,s2A+//i-'cos2B-f/;-6'2cOs20. 



Or, as cosC=sinA, and cosB=cos90'^=0, 



Hence, if w represent the angle GCF, we shall have 



_, // , ^^cogZ^Y-i-c^sin^A 



CoSw=— =:J:: -— 



P Va'cos^A-l-c-'sin^A 



:± 



112 



Va'cos^A-t-c'-'sin^A 



[4G. 



This simple case has been 

 examined in detail, in order to 

 facilitate the conception of the 

 more general one, which will 

 now be attended to. Let a 

 molecular displacement, r, occur 

 in any direction whatever. Let 

 CF, Fig. 61, be the direction of 

 displacement, and let it be as- 

 sumed as the representative of 

 the force developed in that di- 

 rection. As in the former 

 case, if this force be resolved 

 into three component forces in 

 the directions of the several 

 axes, the resistances developed 

 will be /m^cosA, //v^'cosB, 



According to the laws of the 

 composition offerees, these three 

 components are in the relation 

 of the three dimensions of a parallelopipedon, of which the resultant is the 

 diagonal. Let CNN'GH, &c., be this parallelopipedon. CG is the resultant 

 expressing the total resistance in both quantity and direction. But R'^h by 

 the equation of the surface, expresses the total resistance in the direction of the 



Fiff. GL 



