4.0 Professor Hamitton’s Third Supplement 
1 psa"? 730%? /80'y2 : 
SSS a A 2 (H") 
» having the same meaning as in the second number. In effecting the last elimina- 
tion, we have attended to the relations ( ¥*), which give 
oW &Ww (= )'= W's (Se ) 
dy? 822 \ By’82" 
SW ew (wy? TOT Ba. fo Oe 
S22 oe ( Oz 6a" Ne = Ge ) 
WV SW ew \? pe Mo 
a2 Dy -(=.;) =” ea ii 
or (OY éx'dy (1°) 
ow ow wighs W ew = Wy” < 6ar . 
S2'8y' 82/8x! 8x"? By'Sz! — bu’ ” 
BH Se FW a W8Q' 80% 
Sy'sz! Sa’dy’ Sy” Bax! aoe = bo! ” 
DH OY SMOEK, De mea! sa" 
32'S2" ? d/o, = 72 82 dy! ee v Sel pe 
And combining (2°) (F"), we obtain the following formula for #7’, analogous to the 
formula (Z*), which completes the solution of our present problem, because it is 
equivalent to twenty-eight expressions for the twenty-eight partial differential coeffi- 
cients of 7, of the second order, deduced from the coefficients of W ; 
O=0" IV" fe T+ (W-T)8Q-VV 80-28, W .32'-87W + W280! + y'8r' + z'bv'Y8Q y 
te ‘(oa 27) — vhf eae) U 
ety ( 7 Me (ae ah yy 
(ee vy es 5 ( bo" 5% ye 
eA | a ’ (sea? w) ti ( Sy! ag BN «(2 ay 
+25 v! oe As a ~ ye) (eZ yh 
m) 
, 
ha ‘ : OW Bi) TEs Qo as 
+25 = ae or — ae ( 30’ a8 eka (8v' ey er —v (x — a) h (K°) 
And if we denote by 8°7',, the value of the second difféventinl &T assigned by the 
formula (JC *), and determined on the supposition that Z’ has been made, before dif- 
ferentiation, homogeneous of the first dimension with respect to 6, 7, v, and also with 
