99 
wm 
Professor Hamitton’s Third Supplement 
and hence by (.4*) we can deduce already, without any farther differentiation, 
SV 1 ew LP ae by BY by BF 
st ow? Cpe 7 * 5D? Sede 8 Oe Oe Sey? 
a Su SV tv SV ou &&V 
Say = 7 wAaode —255,)3 Syoz 8a Sady | or oy? (G*) 
37 1 Sw See a dy 
By? wo” Vace Boe J? Be Be Orde Or dyes 
observing, in deducing the sixth of these equations (G*), that by the definitions 
(EZ), and by the a of v on, 7, x, we have 
age du .oY ov 
gen =(=) 232 Fp ay it 5 ox (H’) 
The equations (4*) (F"*) (1°) give also 
OV as SW ew o°u 1 &w Ww o2u 
vox’ ie eee =e w” Sod’ Crs tee Ds 
eV aa: Lae REZ LOA 
arey wv" ay Sadr Sadr? wo” Body’ De See 
BY _ SW (SW Bey 1 SW SW ey, 
Sziz) w” SroZ “Sadr * Bade] we” dade We * Be} 
oy Sein ciao Ne ite Se i Se 8 
byde’ ~~ w" sadx" \dadr Oodr a w Srdai “So? “ie a) 
SV 1 &W ew o2u 1 &W sew ov 
SySy wm” Sody Sede * Sa) wo Bye Baad =) ; (P) 
BY gs) SG PIE | WEe yas LO EG Bi 
dydz’ — w" Sodz' \8a8r dadr7 ow” Sr dz’ gry 
SV b&w RK SV 
dzdv’ 8a Orda’ Sr Syd2"’ 
SV ke SK be 
ody’ oo Oxdy' Br: -Bydy' ” 
SV we we eV. 
dzd2’ ~ 8a Sxdz' Sr Sydz! ”? J 
and 
Se awe ly ow ow Saas 2a 
ar Om ta) (< = 2ex)- Age - =i Ge 
eV 1 ey ew ew ew bu Ke 
yi w Seay 7 7 = Ge a wes - eee * * 52 ; ( ) 
SV vey wey ow 
32x da Bzdy | Or(Syax Sx’ 
We have therefore found expressions (G*) (I*) (A*), for eighteen out of the twenty- 
eight partial differential coefficients of VV of the second order; and with respect to 
