'286 Proceedings of Boy al Society of Edinburgh. [sess. 
The values oi a 2 , aa , ... are throughout given side by side 
with those for a 2 , a/3 , .... ; thus — 
2 1 (G~ - m)(G~ - n) - r _ (Q 2 - j/)(G' -p") - 
^ 9900) 9 9 9 9 * 
(G - G' )(G - G" ) (G - G' )(G - G" ) 
At this point “ Problema I.” stands fully solved : one or two 
interesting addenda, however, are given in a concluding section 
{§ 15). From the equations 
Ga = Aa + B/3 + Cy , s = ax + py + yz , 
G 'b = Aa + B/3' + Cy , and s' = ax + /3'y + y'z , 
G"c = Aa" + B/3" + Cy" , s' = a'x + (3"y + y'z , 
Gc& — 
by multiplication and addition* there are obtained 
Ax + By + C z — Gas + G'&s' + G "as " , 
A'x + B y + C'z = Ga's + G'b's + G "c's " , - 
A"x + B "y + C "z = Ga"s + G'b"s + G "c's" ; 
•and then from these by the second part of theorem (0) 
(Ax + By + Gz) 2 + (A'x + B'y +■ G'z) 2 + (Ai'x + B"y + G"z) 2 
9 9 9 9 9 9 
= Gs + G' s' + G" s" , 
which may also be written in the form 
zj a a z z, 2 2 z, ’/ 
lx + my + nz + 2T yz + 2 m'zx + 2 rixy = G s + G' s' + G" s" . 
To this ot course may be appended the derivative from it by the 
substitution of — — ^ , .... for A , .... viz., 
{(B'C"-B"C> + (C'A" - C"A')y + (AB'-A'B» ! 
+ ((B"C -BC")* + (C"A-CA ")y + (A"B -AB")z } 2 
+ {(BC' - B'C )x + (OA' -C'A )y + (AB' -A'B)z } 2 
2 ,2 2 222 999 
= G G" s + G" G s' -f GG' s" . 
* We may formulate for use here the following theorem in modern dress : — 
If | a/3'y" | he an orthogonant , then 
