or tension coinciuiiig wiiu iiie a.^.e^ "i lino ou-i^vv, v^i •.••^ c^^«..v. ».»,.>. . . - j „..., v..„ 



the three principal pressures or tensions above determined, one will be a maximum and anoth( 

 minimum, while that of an intermediate value will be neither, though it satisfies the condit 



460 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 



and eliminating cos a, cos /3, and cos 7 by cross multiplication, we obtain 



{A-p)(B-p){C-p)-D°-iA-p)-EHB-p)-r(C-p) + 2DEF=o. 



If we take the three values of p de'ducible from this equation and substitute them successively 

 in equations (c), those equations combined with (2) will give three distinct systems of values for 

 cos a, cos /3, and cos 7, belonging (as is well-known) to three lines perpendicular to each other. 



Hence, it follows, that there is at every point {P) of a continuous solid mass under extension 

 or compression, a system of three rectangular axes, such, that if the small plane («) at P be so 

 placed that its normal shall coincide with one of those axes the whole resultant action on s shall be 

 normal to it, the tangential action upon it being then equal to zero. These three axes are called 

 the axes of principal pressure or tension with reference to the point P. 



3. M. Cauchy, in the Memoir above referred to, converts equation (l) into the equation 

 to a surface of the second order, by putting 



« cos ^ = ± T , )• cos a = iv, r cos /3 = y, r cos y = z. 



The inverse of the square of any radius vector will manifestly be a measure of the normal 

 action on a small plane through P perpendicular to the radius vector, the axes of principal pressure 

 or tension coinciding with the axes of this surface of the second order. We may remark, that of 



other a 

 itions 



d . p cos d _ ^^ _^^j -p cos ^ ^ j^ .^^ .^^ ^^^^^ ^^_^^ value of p cos S which is represented by the 



da dji 



inverse of the square of the mean axis of the surface, and that mean axis, considered as a parti- 

 cular radius vector, is a maximum with reference to one principal section, and a minimum with 



reference to the other to which it belongs, so that though — ( — ) =0, and --: ( — j = when 



r = mean axis, all the conditions of a maximum or minimum are not satisfied. 



I make these remarks here because a similar mode of geometrical representation may be found 

 useful in explaining the results obtained in the succeeding part of the investigation. 



4. I shall now proceed to investigate the positions of the plane s passing through P, when the 

 tangential action upon it is greatest, i. e. when p sin ^ = max. 



To simplify our formulas, we may here take the axes of principal pressure or tension as the 

 co-ordinate axes. In this case there will be no tangential force on the plane s when it is perpendi- 

 cular to any of these axes, and consequently, we must have 



X» = 0, E = 0, F =0, 

 and, therefore, equations (a) give 



p' = A^ cos' a + B^ CDS' li + C cos' 7, 



and equation (l) gives, 



p cos ^ = A cos' a + B cos' /3 + C cos' 7. 



Hence we have 



ja' sin' S = A- cos' a + B- cos' ;3 -(- C cos' 7 - {A cos" a + B cos' /3 + C cos' 7)', 

 the quantity which is to be made a maximum subject to the condition 



cos' a + cos' /3 + cos' 7=1. 



By virtue of the last equation, we have 

 /)- sin' d = {cos'a+cos- /3 -I- cos' 7) (J' cos^ a + B^ cos' /3+C cos' 7) - {A cos' a + B cos' /3 + C cos' 7)', 



