218 
Proceedings of Royal Society of Edinburgh. [sess. 
“Man kann den Ausdruck A + A' + A" leicht entwickeln, 
und wird ihn dann durch a' a" theilbar finden,” 
there is evidently a foreshadowing of the identity 
\a! b | , \a' c | , \a' d \ 
| a!' V | , | a!' c' | , | a" d ' | 
\a"'b”\ , \a!"c"\, \a'"d"\ 
- a f a"\ab'c"d "'\ . 
The fourth theorem, concerning the tetrahedron enclosed by 
four given planes, 
A + #B + + (a +b x + c y) C , 
A + &B + yC + (a! + V x + d y) C , 
A + aB + ^C + (a" +b"x + c"y) C, 
A + xB + yC + (a"' + b'"x+d"y) C , 
is made dependent on the third. The intersections II, II', II'', IT" 
of the four triads of planes are found to be given by 
q II = (b c' )A + (c a' )B + (a b' )C + (a b c )D , 
i IT = (&' c ") A + (c f a ") B + (a' b") C + (a! V d )D , 
q" n" = (&'V")A + (c" a"')B + 
q'"IL"'= (b'"c )A + (c'"a )B + (a'"b )C + {a'"b'"d")D , 
where 
(bc') = b(c' -c") + Z/(c"-c) + b"(c-c’), 
(ca') = c(a' - a") + c'(a" - a) + c"(a - a') , 
(ab') = a(V -b") + a'(b" - b) + a"(b-b'), 
and ( abc) = a(bd ) + b{ca') + c(ab ') , 
= a(b’c" - b"c') + a\b"c-bc") + a'\bd-b'c). 
Hence, by the third theorem, 
nnirn'" A+A' + A" 
ABC D -qqY4"(Vcl(b''c"’) i 
where now 
A = S'(J8'V" - P"Y), A' = 8"(/3"'y - Py'"), A" = 8"'(P'y" - P"y) , 
