1901--.] Dr Muir on the Theory of Ortliogonants. 
259 
by solving the three equations of the original substitution for cos if / , 
sin if/ cos cf» , sin if/ sin <f> , or by taking the results as already found, 
and verifying them by substituting the values of cosP, sin P 
cos 6 , sin P sin 0 . 
On returning to the main line of investigation, viz., the solution 
of the set of twenty equations, Jacobi unfortunately does not 
proceed with the same fulness of explanation as before the 
interruption. In fact, the values of the remaining sixteen 
unknowns are merely put on record without any indication of the 
mode in which they have been obtained, “ brevitati ut con- 
sulate,” the first four of the sixteen being 
a (ft — G )(ft — G )ft — Gr ) — C (ft — G ) — c' (ft” — G ) — C (ft — G ) + 2 C C C ' 
h (G + G)(G — G^fG^ — Gr ) 
a (ft” — G')(ft' — G )(ft + G f — c (ft + G ) — b"' (ft” — G') — b /L (ft” — G A ) + 2 b"b" e 
_ — 
(G' + G)(G' — G”)(G’ — G'”) 
a"- (d" - G’)(a + G')(a' - G') - c"~(a + G') - b'\d" - G') - b"'\d - G') + 2 V'b'c 
Jc (Or + G)(G' - G")(G' - G'") 
a'"’ _ (a + G')(d - G')(a" - G') - c"'\a + G') - h "'\d - G') - b%T - G') + C l h'b"c"' 
h (G + G)(G — G )(G — G ) 
and the others obtainable therefrom by the change of 
2 ,2 ,,2 ,,,2 
a ) a j a ) a t G, G , G , G , 
into 
P*. f, G , G", G' , G'", 
y » yf , y'" > y "' 1 , G , G'", G", G' , 
— 8 5 — b , — S” , — S ' , — G, — G, G, G”. 
The difficulty of the double sign which appears in every case is 
got over by merely fixing at will the sign of a , /3 , y , S , — the 
reason being that there are rational expressions for 
aft , aa” , aa'” , /3j3' , . . . , yy , . . . , SS' , . . . . 
and indeed also for 
a a' , aa ' , a!’ a" , 
similar to those just given for a 2 , . . . For example, 
*a = b'(a" - G ')(a'" - Gr') - c"b"\a - G') - c"b"(a" - G') - 5V 2 + b"c'c" + b'"c'c " . 
h (G' + G)(G' — G")(G — G”') 
