1901 — .2] Dr Muir on the Theory of Orthogonants. 
285 
2 ,2 
As an alternative to this process for finding a , a , . . . there is 
given another, which in some respects is the more interesting of 
the two. Beginning with a different set of equations, viz., the set 
(l - G 2 )a + rifi + m'y = 0 j 
n'a + (m - G 2 )/3 + Ty = 0 
m'a + T/3 + (n - G 2 )y = 0 ) 
Jacobi drops out the first and finds a : /3 : y , drops out the second 1 
and finds /3 : y : a , drops out the third and finds y : a : /3. Then 
since these three sets of ratios are the same as the three sets 
a 2 : a fi : ay , jS 2 : /3y : /3a , y 2 : ya : y/5 ; and as the expressions 
found proportional to aft , ay in the first set are respectively equal 
to the expressions found proportional to the same unknowns in the 
other sets ; it follows that 
a 2 , a/5, ay 
P, Pi 
O 
T 
are proportional to 
(m - G ){n - G ) - 1 , l m - n (n - G ) , riV - m'(m - G2) , 
(n - G 2 )(Z - G 2 ) - m 2 , mV - /'(/ - G 2 ) ,. 
(I - G 2 )(m - G 2 )'- « 2 ; 
and therefore that 
a 2 aB 
— 9 o 5 or — 9 - 
(rn — G 2 )(n — G") - V I'm - n'(n - G ) 
a 2 + ft 2 + y 2 
or 
(m — G" j )(? 2 > — G“) + (w — G )(Z — G ) + (Z — G )(m — G ) — l' — m' — tz' 
Here, however, the numerator is equal to 1 : and the denominator, 
being obtainable by differentiating 
9 9 2 
- l)(x — m){x - ri) - V (x - 1) - - m) - n'~(x - ??,) - 2 Vmri 
with respect to x, and substituting G 2 for x in the result, must be 
what is obtainable in the same way from 
( x — G")(.q? — G' )(x — G ) 
and therefore must be equal to 
(G 2 -G ,2 )(G"-G //2 ) 
There thus result the same values for a 2 , a/3 , . . . as before. 
