218 UNDULATORY THEORY OF LIGHT. 



squares of wliose radii will be equal to the elasticities in their directions. This 

 surface, therefore, Fresncl denominated the surface of elasticity. * 



The surface, as might be inferred from the principle of its construction, is an 

 ellipsoid, of three unequal axes. 



Now, it is a point iraportaut-to be clearly conceived, that when, in a medium 

 of variable elasticity, the equilibrium of forces is disturbed by a displacement 

 of its molecules in a given direction, the resultant of elastic resistances excited 

 is not generally in the line of the displacement. Were the displacement to take 

 the direction of one of the axes of the surface of elasticity, the resistance would 

 be directly opposed to the disturbance. But suppose it to be in the direction 

 of some oblique radius ; and, to simplify the matter, suppose this radius to be 

 in a plane passing through two of the axes. Let then, in Fig. 60, ADBE be 

 the section, passing through AB and DE, the axes of greatest and least elas- 

 ticity. Let, a molecular disturbance, which we will call 

 r, take place in the direction CF ; and, for fticility of 

 conception, let us take the line FO itself to represent 

 the resistance it encounters in this direction. AU is a, 

 and DC is c. Now, the displacement r, if it took place 

 wholly in the direction of a, would develop a resistance 

 proportional to rd^, or equal to Jra^, J" being a constant. 

 And if it took place wholly in the direction of c, it 

 would develop a different resistance =y)r^. But the 

 amount of displacement in the direction of a is only 

 rcosA. And that in the direction of c is only rcosG. 

 Also, as A=ACF in this case, and 0=90^— ACF, we have rcosC=rsinA. 

 Hence the resistances developed are yz-a^cosA, and frahiuA. Now, the first 

 of these expref^sions being the horizontal component of the resistance (as the 

 figure is drawn) and the second, the vertical, the second divided by the first 

 will e-ive the taneent of the inclination to a, which inclination we will call A'. 



Then •{-J^=4tanA=tanA', [45.] 



y^a^cosA or ^ -" 



which is less than tanA — or the resultant is less inclined to a than FO. 



A graphic method of determining the resultant, both in magnitude and di- 

 rection, is suggested by this formula. TanA' is a fourth proportional to d^, c?, 

 and tanA. balling KG radius (for present purposes) FK=tanA. Draw FR 

 perpendicular to DE, and join IIB. Join AE, bisect it in M, and draw MN 

 perpendicular to AE. With N as a centre, describe the arc EP. Then CE^= 

 c2=AC.GP. And GB2==a^=AG.GB. Or, a^. ^.2. . CB : GP. Draw, there- 

 fore, PQ" parallel to BH, and QO parallel to AB. Draw FG perpendicular to 

 FG, and the radius GO, through 0, to meet it in G. GG is the resultant, and 

 GGA=A'. 



The resultant force consists, then, of two components — one, equal and oppo- 

 site to GF, and the other FG at right angles to it. This latter force deliects 

 the motion of the molecule in FG, and turns it toward the shorter or longer axis, 

 aceordinc-- as the movement is one of condensation or of rarefaction. And there 

 can be no stability of oscillation in this place, in any line which is not parallel 



* This polar equation may be referred to rectangular co-ordinates, by putting x, y, and z for 

 the co-ordinates parallel to a, h, and c, respectively, and substituting the following values: 



R==2-2-f/4-z-. CosA=f-; cosB=:f^; cosC=|. 

 ' -^ K K K 



Whence ^*^=a:H'^-\-lr%f--\-(i-2?. 



Or, (z2+2/'^+:2)2=:a-x2^&2j/2.fc22^ 



which is an equation oi" the fourth degree. 



