bo 
42 Mr Brill, On the Orthogonal System [May 21, 
The first of these equations may be deduced from the other three, 
and may consequently be left out of account. The other three 
must be satisfied independently of the values of the ratios dx:dy:dz. 
This gives rise to the three groups of equations :— 
ae aS ae 
h, oy * h, ey 
LOE ney _ moe _, 
hoz hoz hoz 
een Rs neOe sD 
hoe hoe h, ox 
fom, tee née 
aoe aS us, Wace BP B), 
h, oy h,oy  h, ey y ®) 
LOH Pet AC 
los Wh oe\ males a 2 
fo, mae Tn _ 
h, Ou h, 0a hy 0x | 
fo moe 1m / | 
h, oy h, dy h, oy 
HGS Lp ULE 
h,dz  h,0z h, Oz — J 
These nine equations between the four ratios f:1: m:n, unply 
six relations, which may be shewn to be equivalent to each of the 
groups at the beginning of Art. 9. 
From the first two equations of group (A) we obtain 
Hetil ED et in ae 1 at a€ 
hh, 0x 0y h, oa hi, dy hi oy\ h, Oa) * hh, Ox dy’ 
= 1a, _1ae_ 1 a6 , Lae 
a 7 hh, O(@,y) h,0v) hh, o(a,y)  h, oy’ 
lee piven) weiimion fee 
oF nea ioe Sin he 
Similarly from the first and third equations of the same group we 
should obtain 
iC) ace 1 On A 
at (men | Sinha 
