32 Professor Hamitton’s Third Supplement 
which may be shown to agree with the less simple equations of the same kind in the 
First Supplement, and may be thus summed up, 
VE 2 2 + pi : S (180—v8r)? ar 3: (weds — adv ) (6dr — rs) 
a == = (ote _ sou)” + on x, (ar — ros ) (réu — vér) 
lay ES —rd0)? + 9° (rdu—ver ) (vdc—adv), (K*) 
= 
the mark of variation &” 1 a only to the variables «, 7, v, as & referred only to 
Bs Ys Zo X- 
And the whole system of the twenty-eight expressions for the twenty-eight coeffi- 
cients of JV’, of the second order, may be summed up in this one formula : 
vV" BW + V8O+ B80 480) = EF Ae 8) or mor 
Fe Att 8) 8 EES om 8) 0 20 ro) 2 
Eafe tor DY nt sie) 
Oe og) 8 Zt (w- 8 )-+(a-eZ)t 
+e" (FES -Cst PLE «(ant B)— (ne Py} 
in which the symbols &, 8", are easily understood by what precedes, and in which the 
seven variations 80, Sr, du, da’, dy’, 82, 8x, may be treated as independent of each 
other. 
The formula (/¢*) has an inverse, deduced from (X°*), namely 
ele eo iA ve 2 
vey” =(Ga =? G = ay) 
ol 6Q oan = O 82 2) 
@ Wess = 
+( a 2a. @ oy as © ar) 
pa@har ye 0 5, 9 y) (2 Sr “a®) 
+2(Se 40 veo) es wa) (Say Se © ar) 
sa@her la) yee) Bee) a0 
