274 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the full substitution being 
II 
*7 
2^124 + 
W f2 + 
~ AT 
ll 
8 
’^ 21 £ _j/2E 2 2 _ I \> , 
■p i+ It v r- + 
2Iw 
~AT^ 
II 
CO 
8 
II 
We may add, that had the relation of the reverse substitution to this not 
been already known it would have been evident from the set of equations 
which here produce both. The result reached is that the n 2 coefficients 
a n , . . . , a nn for the transformation of rectangular co-ordinates can be 
expressed rationally in terms of Jnfn — 1) arbitrary quantities l rs 
satisfying the conditions l rs = — l sr , l rr = 1 by forming the determinant 
I bib ‘2 • • • l nn j V or A say, and thereafter the adjugate determinant 
I R 11 R 22 • • • b nn | , cmd taking 
a 
rs 
a rr 
By way of illustration Cayley works out the cases where n = 3 and 
where n = 4. For n = S he begins with three quantities 
v — /x 
A 
and obtains the substitution-coefficients 
1 + A 2 — /Lt 2 — V 2 
2(A fx 4 - v) 
2(vA — /x ) 
1 + A 2 + /A 2 + V 2 
1 + A 2 4- /A 2 + V 2 
1 +A 2 + /a 2 + v 2 
2 (A/A - v) 
1+/a 2 -v 2 - A 2 
2 (fxv + A) 
1 + A 2 + /A 2 + V 2 
1 +A 2 + /* 2 + v 2 
1 +A 2 + /a 2 + v 2 
2(vA + /a) 
2 (fiv — A) 
l+A 2 + ya 2 + v 2 
1 + A 2 + /i 2 + v 2 
l + A 2 + /U , 2 + v 2 
remarking, in passing, on Rodrigues’ introduction of them (but on this point 
see Euler’s memoir of 1770) and on their connection with the theory of 
quaternions. For n = 4< he begins with the six arbitrary quantities 
a b c 
~h g 
-f 
and obtains for the substitution-coefficients the following quantities all 
divided by A 
