201 

 Methods in Solid Analytics. 



By Arthur S. Hathaway. 



Define the "vector" [A, k, m] as the carrier of the point (x, y,z) = P, to the 

 point (c -\- h, y -\- k, z-)-m) = Q, and show that the distance and direction cosines 

 of the displacement PQ are given by functions of the vector called its tensor and 



unit, T\h, k, m~] = -* /(^ + k" -\- rnr ) = n, U\h, k, m] = [h/n, k/n, mhi\. 



Interpret the sum [A, &, m] + [^j &', " (/ ] — I7< + A', A + //, m -f- ?>/] as a 

 resultant displacement, PQ -f- QP = PP, and the product ».[A, &, m] = [«A, ?iA-, ?im], 

 as a repetition of the displacement. 



Define the linear functions of q=[x, y, z] as the "scalars" or "vectors" whose 

 values or components are linear homogeneous functions of the components of q, 

 such as ax -f- by -j- cz, etc. Hence, for a linear function Fq, F(q -)-/•)= Fq -f P/-, 

 nP<7 = F(nq). 



Hence, for a bi-linear function Fqr, F(aq -f- a / <7 / , 6/" -f frV) = abFqr -\- 

 ab'Fqr' -f a'bFq'r + a'VFq'r' '. 



A special scalar and vector bilinear function of q — \_x, y, z\ q / ' = [a^, y', z'] 



are defined. 



Sqq / = xx'-\- yy'-f zzf^Sq'q. 



Vqq'= \_yzf — zy', zx' — x/, xy / — yxf~\ = — Vq'q. 



If be the angle between the displacements q, q / , these functions are inter- 

 preted as, 



Sqq'=Tq.Tq'. cos9. TVqq' = Tq . Tq' '. sinG; and Vqq' is a displace- 

 ment perpendicular to both q and q', in the same sense as the axis OZ is per- 

 pendicular to OX and OY, i. e., on one side or the other of the plane XOY. 



The use of this material is illustrated in the following examples: 



A=(2, 3,-1), P=(3 ; 5, 1), G = (8, 5, 2), P=(5, 7, 11). 



1. Find the lengths and direction cosines of AB, AC, AD. 



Ans. TAB = 3, UAB=[l f, f], etc. 



2. Find cosPAC. Ans. SVABVAC—\\- 



3. Find area of JPG' and volume of ABCD. 



Ans. J TFAPAC'= | 185, £ SADVABAC— — 13. 



4. Find the cosine of the diedral angle C — AP — D. 

 Ans. SUVABACUVABAD — 



37 V 10 

 5. Find the sine of the angle between AD and the plane ABC. 



Ans. SUADUVABAC=- ir_- 



Vl85 



