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1901-2.] Dr Muir on the Theory of Orthogonants, 
l=m = n=a=b = c = i , 
a ,b , c = cos X , cos /x , cos v 
He then from the second set of six equations forms three groups 
La • a + Ma • a' + lS T a • a = A 
La -p + Ma'./T + Na"-/5" = C 
La • y + Ma' • y + Na" • y 
L/5 • a + M/5' • a' + N/5" • a" == c , 
L p-p + M/5' • /5' 4- N/5" • /5" = B , 
L/5 • y + M/5' • y + N/5" • y" = a , 
Ly • a 4- My' • a' 4- Ny" • a" = 5 , j 
Ly./5 + My' • (3' + Ny"-f3" = a, V 
Ly • y + My' • y + Ny" • y" = C , ) 
and solves the first group for La, Ma', Na"; the second for L/5 y 
M/5', N/5"; and the third for Ly , My', Ny"; the results being 
La = (/5'y" - /5"y')A + (y'a " — y"a')c 4- (a'/5" — a"/5 ')& 
Ma' = (/5"y - /5y" )A 4- (y"a — ya" )c + (a"/5 - a/5" )6 
Na" = (/5y' — /5'y )A 4- (ya' - y'a )c 4- (a/5' — a'/5 )& 
L/5 = (/5'y" — /5"y')c + (y'a" — y"a')B 4- (a'/5" - a (3')a ^ 
M/5' = (/5"y -/3y")c 4- (y"a - ya" )B + (a"/5 - a/3" )a > 
N/5" = (/5y' - /5'y )c + (ya' -y'a )B 4- (a/5' -a'/3)a) 
Ly = (/5'y" — /5"y')& 4* (y'a" — y"d)a 4- (a'/3" — a"/5')C j 
My' = (/5"y-/5y")6 4- (y"a -ya")a 4- (a"/5 - a/5" )C V 
Ny" = (/5y -py)b 4- (ya' - y'a )a 4- (a/5' - a'/5 )C ) 
Making the substitution above referred to he derives the corre- 
sponding results which are obtainable from the first set of six, viz. : 
A • a = (/5'y" - /5"y') + (y'a" - y"a') cos v 4- (a'/5" - a"/5') cos /x | 
A • a' = (/5"y - f3y" ) 4- (y"a - ya" ) cos y 4- (a"/5 - a/5" ) cos /x > 
A • a" = (/5y' - /5'y ) 4- (ya' - y'a ) COS v 4- (a/5' -a /5 ) COS /X ) 
A • (3 = (/5'y" - /5"y') cos v 4- (y'a" - y"a') 4- (a'/5" - a"/5') cos X ^ 
A • /5' = (/5"y - /5y" ) cos v 4- (y"a - ya" ) 4- (a"/5 - a/5" ) cos X > 
A • /5" = (/5y' - /5'y ) cos v 4- (ya' - y'a ) 4- (a/5' - a'/5 ) cos X ) 
A • y = (/5'y" - /5"y') cos /x 4- (y'a" - y"a') cos X 4- (a'/5" - a"/5') 
A • y = (/5"y - /5y" ) cos /x -I- (y"a — ya" ) COS X 4- (a" /5 — a/5" ) 
A • y" = (/5y' - /5'y ) COS /x 4- (ya' - y'a ) COS X 4- (a/3' - a'/5 ) 
