118 
Mr. Whewell on calculating 
±\ _i_ 
p I nc 
and its symbol is {p;p — r\p — c^. 
In the same manner the angle y gives a plane (q—r;q; q—p) 
and the angle z a plane (r — q\r — ; ^)- 
Hence the co-existent planes are 
(P I q I r),(p ;p — r ; p — q),(q — r ; q ; q —p), {r—q; r—p ; r). 
These four planes would truncate symmetrically the four 
faces of one of the pyramids which compose the octahedron, 
and planes parallel to them would truncate similarly the 
planes of the other pyramid. 
38. Prop. To find the portions cut from the edges of the 
octahedron by the plane {p; q;r). 
Let the plane P, Q, R, fig. 16 and 18, meet DK, DE, DF, 
DH in S, T, U, V. Draw QL parallel to DE. Then 
DS = DP = DP . DP . 
AP ha q a . 
QL = 
A.q^ = kb.:^r=kc,hL = Aq^=kb.^ = ka-,-ph^{h 
k) a 
DT = DP .# = DP^t-^ 
PL (ft — ft) a 
= DP.--L. . 1 
q — p a 
Similarly DV and DU would be DP • ^ and DP . 
p 
r — p 
b 
a 
Hence DS, DT, DU, DV are as -1- . 6, — c. —I— b, -L c. 
q q—p f — P r 
Hence for the four co-existent planes the edges cut off are 
respectively as p c__ b ^ 
q ’ q —p’ r — q^ r ’ 
h c b c 
r — jj’ r’ q^ q — p"* 
b c b c 
q * r ^ r — p^ q<— p^ 
b c b c 
r — p' q-p' ~q.' ~r' 
