30 Professor Hamitton’s Third Supplement 
attending to (G’) and (/V’*). 
If we substitute these values of the multipliers, in the 
seven equations ( Y*), we may decompose each of those equations into seven others, 
by treating the seven variations 8o, dr, dv, d4’, &y/, dz’, 8x, as independent; and thus 
obtain forty-nine equations of the first degree, of which however only twenty-eight 
are distinct, for the determination of the twenty-eight partial differential coefficients 
of the second order, of JV considered as a function of o, 7, v, a’, y', 2, x, which 
relatively to «, 7, v, is homogeneous of the first dimension: the corresponding coeff- 
cients of the first order being determined by the seven equations (G@) (W*) (W*). 
Instead of calculating in this manner the coefficients of /V of the second order, 
by eliminating between the equations into which the system ( Y*) may be decomposed, 
it is simpler to eliminate between the equations ( Y*) themselves, and thus to obtain 
expressions for the variations 
of the coefficients of the first order, from which expressions the coefficients of the 
second order will then immediately result. 
first equations ( Y*), in order to get expressions for the three variations 
ow ow OW 
Se oe 
we find, after some symmetric reductions, 
8 
Eliminating, therefore, between the three 
a V8 tr rhage) 4 (FE) (FE) F 
O97 + a(t SA c (8 5r) =» (Be en | 
(es vow) 4 + (%-8F)-o(%-eZ)ts 
= vas (veo Ee) fo (2) - + (w-2Z)b 
837 + (ve se) {2 (85 )—» (2 vz) @) 
oy? aoa, — ie) An Be )— <8) 
se = - vees— («oe - or) Sv ( —¥5r)— + (m— ar) t 
Seville wo) f « ( ~ 85) — » (@- 95) b 
+7 (0 ge team) Lr 8E)—o (88) fo 
in which, 
