CIRCULAR SECTIONS OF SURFACE OF ELASTICITY. 



OC 



21 



in the same straight line. And if we sup])oso these refracted waves to be com- 

 pounded of the infinitely numerous ehimeutary waves which may be imagined 

 to originate in the line of inters(!ction, each resultant refracted wav(! front will 

 be a common tangent plane to all the elcminitary waves of its own kind thus 

 generated. 



Though the planes of vibration of the two re.'fracted rays are originally per- 

 pendicular to each other, yet the taking of different velocities slightly modifies 

 this relation. The change is hardly sufficient to be sensible. 



Ther(; are two sections of the surface of elas- 

 ticity which are circles. Let, for example, in the 

 figure annexed, the axes of elasticity be; OX, OY, 

 OZ, and let the dotted lines represent the contour 

 of one-eighth of the surface of elasticity; OP 

 being =«, 0A=;5, and OE=c. Upon the same 

 axes, with OA=b, as radius, let there be con- 

 structed a corresponding portion of tin? surface 

 ,of a sphere, AC13; in which AC, A13, BO are 

 quadrants. Bince a is tlu; largest, and c the least 

 axis, the ellipsoidal and spherical surfaces must 

 cut each other somewhere between B and C. 

 They will also toucli at A. Let one point of the 

 intersection be at R, and through 11 pass a plane, OAU, intersecting the spherical 

 surface in All. In this, take any point, as N, and draw through it the quadrants 

 BNS, CNQ. Considering N as a point of the surface of the sphere, the radius 

 ON=i, and we have 



/;3=i\-Os2A+Z-2cOs2B+Z»2cOs2C. 



Considering it as in the surface of the spluiroid, ON=R, and 



R2=a2cos2A+Z'-cos-B+6-2cos2C. 

 If both these suppositions are true R-=Z*^; whence we deduce 



(^2— Z,2)C0S2A=(^.-— f-)c0s2C. 



If a be put for the arc CR, the inclination of the plane, ANR, to the axis a, 

 then, in the triangle CNR, we have 



Cos2CN=cos2A=cos2NR cos2CR=sin=^B cos^a. 



And, in the triangle BNR, 



Cos2BN=cos2C=cos2NR cos»BR=sin2B sin^a. 



Substituting these values for cos^A and cos^C in the foregoing, and dropping 

 the common factor, sin^B, 



{a^ — i-)cos^a=(i2 — t^Jsin^a 



c?—l? sin^a , 2 r» + i W~lj' rial 



And 75 A= — 5-=tan2a. Or, tana=i: / _, [48.] 



the double sign indicating two positions for the section, one in the first, and tho 



