412 RICE 



ART. K 



minors of (12) is equal to the sum of the squares of the ratios of 

 enlargement of three bounded plane surfaces, respectively- 

 parallel to the three principal planes of the strain. Of course 

 the sum of the squares of the nine Crs coefficients is equal to the 

 sum of the squares of the three principal ratios of elongation. 

 The interpretation of these results in terms of ratios of en- 

 largement is of some importance. Equation (13), which is 

 really the third equation of (11), is an especially useful result and 

 is involved in Gibbs' equation [464]. The first equation of (11) 

 is perhaps the least important of the three for our purpose, but 

 the second result in the form of equation (15) plays a part at one 

 or two points of Gibbs' treatment, e.g., at equation [463] and still 

 earlier on pages 192, 193. It will be well to pause a moment 

 to consider the geometrical significance of the nine minor 

 determinants £"11, £"12, etc. To this end let us imagine a triangle 

 P'Qi'Qi in the unstrained state such that the local coordinates 

 of Qi, Q2, with reference to the local axes at P', are ^i, r;/, n' 

 and y, 772', ^2- After the strain the triangle will assume the 

 position P"Qi"Q2". If ki", Vi", Ti" and ^2", V2", h" are the co- 

 ordinates of Qi" and Q2" with reference to local axes at P" 

 parallel to the original axes we have by (3) the following rela- 

 tions: 



ki" = en^i' + enm' + eM, y = eM + 612^72' + ei3f2',] 



Vl" = €21^/ + 6227?/ + 623^/, 772" = 621^2' + e22r?2' + ^23^2', \ (16) 



fi" = 631^/ + 63217/ + e33fi', h" = ez^y + e32i?2' + 633^2'.] 



Denote the area of the triangle P'Qi'Qi by K' and that of 

 P"Qi"Q2" by K". The projection of the triangle P'Qx'Qi' on 

 the local plane of reference perpendicular to the axis of x' is a 

 triangle whose corners have the 77, f coordinates 0, 0; 7?/, f/; 

 772', ^2'. By a well known rule its area is livi^i — f]2^i), and 

 similar expressions hold for other projections. Now the area of 

 a projection is equal to the product of the projected area and the 

 cosine of the angle between the original plane and the plane of 

 the projection, which is the angle between the normals to the 

 planes. So if a, /3', 7' are the direction cosines of the normal 



