On Systems of Rays. 57 
82’, dy, may be treated as independent in (/?’), if we assign a proper value to od, as 
a linear function of these seven variations ; so that we may deduce from (£') the 
seven following equations, 
As oF 03 HO aus an ] 
AS Be =A? get gy ut BMS 
as =28 ae au +o d; (S?) 
AS was ; AB =a ; nae = du; 
Ad aes 
of which each may again be decomposed into seven others. But of the forty-nine 
expressions thus obtained for the changes of the twenty-eight coefficients of V of the 
second order, only twenty-eight expressions are distinct ; and these involve seven 
multipliers as yet unknown, namely, the seven partial differential coefficients of 2 : 
however we can determine these seven multipliers, and the twenty-eight coefficients 
of FV, of the second order, by introducing the seven additional equations obtained by 
differentiating the partial differential equation (#”), with respect to cy z 2’ y/ 2 x. 
The differential of the equation (#”), is 
6Q2 éQ, OQ 
2 Ve 2 Vy 2 V, 
_ 60, 38 2, OQs 3° 2 , 6Q, 9° 2 , 6 on + oy + ohid Geeks (T’) 
Dee SG ON etree ine atnisien' Bie By cone 
and this, when combined with the three first equations (S*), conducts to the following 
formula, 
om 3 a ian +o ee + an ay + be + Se by 
th (Geet Oy tae 2) 
tu (asin By Th Be) 
+0. (@ a ete =) 5 (U’) 
which resolves itself into seven separate equations, sufficient to determine the seven 
multipliers 
AB A A A A 
dz’ dy’ 827 Be ? B/ ? S’” dy° 
VOL. XVII. Q 
