280 REPORT—1862. 
independent variables. Should the three equations (27) be satisfied, the 
expression (31) will be simplified, becoming 
lu dv dw dw dv du. dw 
ges Po ge aw sca Ne?" (lea ase {eee 
eae ages igh @ dz * te a(ateyth q ata) 
dv, du 
+1. (G45) eco ke ee 
where T, denotes T,. or T_,, and similarly for T,, T.. 
The general expressions for the tensions resulting from Cauchy’s method 
are written at length in the equations numbered 17 and 18, pp. 133, 134 of 
the 4th volume of his ‘ Exercices de Mathématiques,’ where the normal and 
tangential tensions, referred to surfaces in the actual state of the medium, 
are denoted by A, B, C, D, E, F. These expressions contain 21 arbitrary 
constants, of which six, A, 33, €, B, €, Jf, denote the tensions in the state of 
equilibrium. If these be for the present omitted, the remaining terms will 
be wholly small quantities of the first order, and therefore the tensions may 
be supposed to be referred to a unit of surface in the actual, or in the 
undisturbed state of the medium indifferently. On substituting now for 
1 ge Pp; i he A T,, T, in (82) the remaining parts of A, B, C, D, E, F (observing 
that the £, n, in Cauchy’s notation are the same as u, v, w), it will be seen 
that the right-hand member of the equation is a perfect differential, integrable 
at once by inspection, and giving 
du\? dv\? dw\? dv, dw\? , g dv dw 
— 21 (FE) +0(T) tN (e) +P (EtG) aga 
dw , du, 5 dw du | | du dv\? , odu ot 
+01 (G+5 +20o | +R (Gta 1a 
du (dv . dw dw , du\ {du . dv 
4201 eet ay) tet ae) (ay tae) | 
,{ dvfdw . du du. dv\ (dv . dw 
ro \z (ate) H(ata) (ets) | ) 
dw (du. dv dv .dw\ f/dw_. dw 
D) Wee sikh ae Ha 
ae { dz (+a) +(at 7) dx ii =) } 
du fdw . du dv {du . du dw {dv . dw 
NV on es Wie ee SE Oa ey 
Ea dx Get %) een dy (Gta)t e dz \dz* dy 
du (du , dv\ , op du (dv , Wwy , on dw (dw , du 
ae (ay tida) elt gg a ee 
the arbitrary constant being omitted as unnecessary. We see that this is a 
homogeneous function of the second degree of the six quantities (24) and (25), 
but not the most general function of that nature, containing only 15 instead 
of 21 arbitrary constants. 
Let us now form the part of the expression for ¢ involving the constants 
which express the pressures in the state of equilibrium. It will be convenient 
to effect the requisite transformation in the expressions for the tensions by two 
steps, first referring them to surfaces of the actual extent, but in the original 
position, and then to surfaces in the original state altogether. 
Let P’,, T',., &c. denote the tensions estimated with reference to the actual 
extent but original direction of a surface, so that P’, dS, for instance, denotes 
the component, in a direction parallel to the axis of w, of the tension on an 
