98 
Mr. Whewell on calculating 
1 1 Prop Knowing the lateral angles made, at the termi- 
nal edges, by the planes of any bipyramidal dodecahedron 
to find the symbols. 
If we have planes (/>, q, r) they will generally form a bipy- 
ramidal dodecahedron, and the six angles at the edges of 
each pyramid will be alternately greater and less. If p, q,r 
be the order of magnitude of the indices, p being the great- 
est, the order of the faces will be that represented in fig. 
(see hereafter the section on the arrangement of faces). 
Hence faces occur in the order (/> ; q; r) {q Ip', r) (^r; p; q) 
&c. : and if 9 be the angle of the two first, and 9 ' of the next, 
we shall have 
zpq — (P®+ ?*+ 2 r + 2 y r) cos. 
COS. 9 = 
COS. O' = 
— 2, (^p q p r q r) cos. « 
z qr p * — (q‘* r* 2 p q z p r) cos. * 
P* + — 2{pq-\-pr + qr) cos. « 
from which equations we have to determine q and r in terms 
of p. 
To eliminate in these equations would lead to expressions 
of four dimensions, and it will generally be simpler to find 
q and r by trial. If we assume for p any number, as 12 ; 
q and r, which generally bear to it very simple ratios, will 
in most cases be whole numbers, and may be found by a few 
trials. And if the ratios of q and r to p involve quantities 
which are not divisors of 12, still the trials made on this 
supposition will indicate nearly the values of q and r ; and by 
trying other values for />, we may obtain them accurately. 
If two of the indices, as q, r be negative ; the order of 
the faces will be (^ ; — r ; — ^) ( — '* ; p\ — q){ — q '■> P \ 
&c. and the rest of the process will be the same as before. 
12. Prop. Knowing the angles made by any plane with 
two primary planes, to find its symbol. 
