On Systems of Rays. 4S 
7 tee 
We have also, in this case of a single uniform medium, 
Vao(e—2')+ry—y)+v@-2), — (¥) 
and therefore, by (D') (#') (U°), 
ANG +rYy +21 ' (2) 
the last of which results may be verified by observing that the general expression for 
the auxiliary function 7’ may be put under the form 
3V ar SSS er ae a 
DS Crag tt Wig at 2 axe Ht Dee 7 A, oy +2 aan VU, (AS) 
so that 7’ vanishes when V is homogeneous of the first dimension with respect to the 
six extreme co-ordinates. The formul of the last number, for the partial differen- 
tial coefficients of 7, all fail in this case of a single uniform medium, for the reason 
already assigned ; but we may consider all these coefficients of J’ as vanishing, like 
T itself: we may however give any other values to these coefficients which when 
combined with the relations betwen the variables will make the variations of TZ’ vanish. 
The coefficients of W may be obtained by differentiating the expression (Z°), which 
is of the homogeneous form that we have already found it convenient to adopt ; they 
are, for the first two orders, included in the two following formule, 
OV =27 80 + Yer +2/8u + c6u' + Toy + vez’, Be 
& W = Badr’ + Q87dy/ aig Wvd-z', ( ) 
and they vanish for orders higher than the second. And the coefficients of V7, of the 
two first orders, may be deduced from those of JV by the formule of the eighth 
number, which are not vitiated by the existence of the relations (U°), because those 
relations do not affect the variables that enter into the composition of VY and JV. 
The variation of V, of the first order, is 
oV’ =o(d4 —82") + + (dy—8y’) + (82 —82') — ye 7 8x3 (C*) 
and that of the second order is given by the following eqiitiond disthced from (0*), 
(W"), (B), 
Cre) FQ 0) +8 ue FQ ‘a 10) y FQ FO =-y << 
302 3s Oaor 2 Sue éréu + 32 da" Ge o+r+u 
627 . ; 3 
9005, ag _p2% een ras 
VOL. XVII. 
