24 Professor Hamriton’s Third Supplement 
oy oy Ww & 8M) eV Sy  8W_y eV : 
ay? - xe dy? ( ) oxvdyx ( orey orox ) oyex : hee 
And if we would generalize the twenty-eight expressions (G*) (I*) (¢*) (L*) CZ") 
(P*), so as to render them independent of the particular supposition, that //” has 
“B0dx  dadx 
been made, by a previous elimination of v, a function involving only the six indepen- 
dent variables «, 7, x, vy, #, we may do so by suitably generalising fifteen out of 
the twenty-one coefficients of JV, of the second order, which result from the 
foregoing suppositions ; that is by leaving unchanged the six that are formed by dif- 
ferentiating only with aeapect to x’, y', 2, but changing , » &c. to the following more 
= &e. 3 
general expressions ee 
a2 
Eo. as Pench be ee dv? OV &v | 7} 
~ 8cdu oc dv? =) du daz ” 
eas) 24 a i Cow oy ow One Hess OW & “U S 
O72 me Ore oréu. or éu? or pun 8 2 
ar) 4 ow 129 oW bu oe oW &v ‘ 
ox” Pee oe oxou ex +3, ov dx? y 
Es a OV ee Ow ou MV Se eW dv bu OW &u é 
= C= ae oréu Se 2 Ne Sun One sue Oo: io 8a6r ’ 
ia |= be ad eure SW bu eee L eur SW &u , 
b08x Ss Perms ae, dy T Sy ba 5x ts oabx 4 
[ 217 ]= SW a CW Ba, SW bu ee Se ou OV &u 
drdy 4 — dxdu or ESS x bu? or By eee drdy 7 (Q’) 
(ee eee BW by eH 4 _ 8h BW bv 
Sobel Bebe + par ao Doar l=aer ther &? 
[= |= BS 74 és OW bu, 1s a OW OW bu. 
Body’ I Scby’ |) Buby 30° LBrdy I~ Srby * Buby! 8? 
OW |= 1 yee =; CW are ee CW SW bu 
Sie ess 7 Soave Sreea | Suber? 
eee ev ew - 
Wel Te Tee 
a aa 
5x5 1 yay * Bvdy’ By? 
ee j= os 
bx oz =P 5x62" a dudz' by" J 
obtained by differentiating the three corresponding expressions of the first order, 
ov ov oWw ou. - OW bu_ ol es Oe év , 
ails: Suan? = “|= Ine Ses [5 l= Bese? ae 
which are to be substituted in (B*), in as of 
SV OW BW 
pee ine 
