1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
255 
a multiple of the original denominator p must be obtained, it 
follows that the expression 
G' (a + a cos if/ + a" sin if/ cos <f> + a" sin if/ sin <£) 2 
+ G" (P + P' cos i J/ + P" sin if/ cos <j> + P'" sin if/ sin <£) 2 
+ G'" (y + y cos if/ + y" sin i ft cos cj> + y" sin if/ sin <£) 2 
+ G (8 + 8' cos if/ + 8" sin if/ cos <f> + 8"' sin if/ sin cf > ) 2 
must also he a multiple of p. Putting it equal to lcp , and equalising 
the coefficients, we obtain another set of ten equations 
G r a 2 
+ 
G"p 2 
+ 
n'" 2 
G Y 
+ 
GS 2 
= 
ah , 
2 
Ga 
+ 
G"P' 2 
+ 
G"V 2 
+ 
G8' 2 
= 
ah , 
rv " 2 
G a 
+ 
G"p" 2 
+ 
r\ /// "2 
G y 
+ 
G8" 2 
= 
ah , 
rv ///2 
G a 
+ 
G "P"' 2 
+ 
r\rn rn 2 
G y 
+ 
G8'" 2 
= 
a"'h , 
G^a^ 
+ 
G "pp' 
+ 
G""yy 
+ 
G88' 
= 
b'h, 
Gr'aa" 
+ 
G"pp" 
+ 
G-'yy" 
+ 
G88" 
= 
b"h , 
G'aa" 
+ 
G "pp"' 
+ 
G'W" 
+ 
G88'" 
= 
b'"h, 
G'a"a"" 
+ 
G "p"p"' 
+ 
G"yy" 
+ 
G8"8'" 
= 
e'h, 
G'aa' 
+ 
G"P'"P' 
+ 
r'yftr fff t 
G y y 
+ 
G8'"8' 
= 
c"h, 
G'aa 
+ 
G"P'P" 
+ 
G"’y'y 
+ 
G8'8" 
= 
e'h . 
We have thus a score of equations from which to determine the 
score of unknowns, a , p , y , 8 , a , . . . , G , G' , G" , G"'. 
Prom this point onward the procedure closely follows that of the 
preceding paper. 
Noting that the specialising substitution 
-G = G' = G" = G'" = 1,\ 
- a = a = a" = a" = 1,1 
V = b" = V" = 0,j 
c = c = c"' I 0,) 
changes the second set of ten equations into the first, he confines 
himself at the outset to the second set. From this four sets of 
equations 
are 
selected, 
e.g. 
, the set 
a.’G a 
+ 
P-G"p 
+ 
y-G y 
+ 
m 
= ah , 
a *G a 
+ 
P\G"P 
+ 
y'-G"'y 
+ 
S'-GS 
- b'h, 
cL'G’a 
+ 
P"<G"p 
+ 
y"-G'"y 
+ 
8".GS 
= b"h, 
m r ^ / 
a -vjt a 
+ 
pr-Grp 
+ 
y'"-G'"y 
+ 
<5"'-GS 
= b'"h, 
