two Planes, &c. referred to three oblique Co-ordinates. 
199 
Also 4x + By + Cz =m, A'x + Bly + C'z =m, being the equations 
to the two planes, whose angle (@) is required, if we put for x, y, <, 
their values from (1) these ee become 
da Bb, Ce 
areal 
Be Bb +B) nt 
r+ 
Cr Bw. 
Ae BY, ©) (ta BE, 
ee 
S) nt — 
e 
en owe 
Hence the value rs cos. @ will be given by the formula 
PP'+ QQ'+ RR’ 
= é= 
cos. VAG PE + VY (P* + OF + R?) (B° + Oe Rey REY? 
P, Q, R, P’, Q’, R’, being the coefticients in equations (4). 
By developing the numerator i + QQ’ + RR’ we obtain 
a aa + (AB'+4'B) “24 (404 A'0) 5 7 4 (BC'+B'C) 5° 
+ ae as aes oe + (AB’ + A’B) fe se 905 7 
ae ot +(AB’ +A’ BA (AC +4 ace = a ¢'+(BC4 B'C Wy 
And reducing, by equations (2) and (3), this becomes 
Se ABAD) con. ACHA'O oo5,_ Hees ©). 
The denominator of — cos. 6 being transformed in the same 
manner, we shall find for P* + Q* + R* the expression 
Aa 
d2 
A 2 a’ 
a 
Azar 
d? 
which, as before, is 
S = 
Bb? Cc =2ABab 2ACac , 2BCbhc 
Aa ea ea F 
By 
sisal + &e. 
2Bl2 
a z + &e. 
e 
reduced to 
Cc? Bees re oS 2AC ee . 
+ Fi Gale a df aipn ata 
